Ba y esian inference
of inhomogeneous p oin t pro cess mo dels
Metho d ological adv ances and mo delling
of neuronal spiking data
v orgelegt v on
Master of Science
Christian Donner
OR CID: 0000-0002-4499-2895
v on der F akultät IV – Elektrotec hnik und Informatik
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des ak ademisc hen Grades
Doktor der Naturwissensc haften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr. Georgios Smaragdakis
Gutac h ter: Prof. Dr. Manfred Opp er
Gutac h ter: Prof. Dr. Guido Sanguinetti
Gutac h ter: Prof. Dr. Jak ob Mac k e
T ag der wissensc haftlic hen Aussprac he: 21. F ebruar 2019
Berlin 2019

A c kno wledgemen ts
I dedicate the first lines of m y PhD thesis to explain why this is the only section I write in the first
p erson singular.
I w ant to express m y sincere gratitude to Prof. Manfred Opp er, who has not only b een m y PhD
sup ervisor, but b ecame a men tor to me in the past three years. Without the man y hours of
inspirational con versations, the man y advices and ideas this thesis in its presen t form w ould not
ha v e b een p ossible.
My sp ecial thanks go to Josef, and Hideaki. The vivid discussions and collab oration with eac h of
them alw a ys ended up in in teresting pro jects, b ecause the common scien tific interest quic kly turned
in to friendship. I also thank Hideaki for giving me the great opp ortunit y to come to his lab in
Ky oto for one month.
I am grateful to the Group of Artificial In telligence, namely Andreas, Burak, Cordula, Dimitra,
Florian, Ludo vica, and Theo for many in teresting discussions, Christmas parties and the flo w ers
at ev ery birthday . In particular, I thank Noa for bringing the Póly a–Gamma to the group and
pro ofreading this thesis in the final stages.
I am obliged to the Bernstein Cen ter for Computational Neuroscience Berlin and the Graduierten-
lolleg GRK1589/2 “Sensory Computation and Neural Systems” for their financial and formativ e
supp ort, whic h gav e me all the guidance and freedom a PhD studen t can wish for. I also thank
them for the tra v el supp ort to go to conferences, summer sc ho ols, and retreats. F urthermore, I was
kindly supp orted financially b y the Deutsc he F orsc h ungsgemeinsc haft through the gran t CR C 1294
for t wo mon ths.
F or giving me the opp ortunit y to presen t m y w ork regularly , and the honest, constructive feedbac k
I am grateful to Prof. Klaus Obermay er and the Neural Information Pro cessing Group. I thank
F ranzi for pro ofreading the thesis and for the man y coffee breaks in the afterno on, that help ed to
mobilise the concen tration for the final hours of the day .
Man y thanks to Lara, Greg, and Erik – the studen ts I had the pleasure to co-sup ervised. During
their mon ths in our group I surely learnt as m uc h from them as they did from me.
Finally , I thank m y family for their strong supp ort despite m y troubles to explain what I w as doing.
The last lines I w an t to dev ote to Rob erta: Y ou ha v e all m y gratitude for y our strong encouragemen t,
patience, and lo ve! The past y ears would ha v e b een difficult without y ou.
C.D.
Berlin, 18 th of De c emb er 2018
i

Abstract
Arriv al times of airplanes, p ositions of car acciden ts or astronomical ob jects in space, lo cations of
ecological crisis, spik e times of neurons, etc. are all data that surround us and can b e view ed as
realisations of p oin t pro cesses. No w ada ys, the mo delling of these data b ecomes increasingly more
imp ortan t, when we attempt to dra w meaningful conclusions from this ev er expanding amount
of data. Mo dels describing the statistics of p oin t pro cess data ha ve been prop osed in the past.
Ho w ev er, to extract the mo del parameters giv en the observ ation of p oin t pro cess data, is generally
c hallenging. P oin t pro cess lik eliho o ds of the observ ed data giv en the mo del parameters are difficult
to deal with in practice b ecause of their functional form. F or Ba y esian inference, where w e aim at a
tractable p osterior distribution o v er the mo del parameters giv en the data, the task is ev en more
demanding.
In the first part of this thesis w e fo cus on a sp ecific mo del class for p oin t pro cess data. The
cen tral ob ject of p oin t pro cess mo dels is the non–negativ e in tensit y function, whic h determines
the lik eliho o d of registering an ev ent at an y giv en p osition in the observ ed space. T o enforce
non–negativit y , p oint process mo dels ha v e b een prop osed, where the in tensity function depends
non–linearly on the mo del parameters via a scaled sigmoidal link function. By the augmen tation
of laten t v ariables w e sho w, that the lik eliho o d of this mo del class can b e rendered in to a no v el
fa v ourable form enabling efficien t and fast Ba y esian inference sc hemes for a tractable p osterior o v er
the mo del parameters. W e utilise this new augmen ted form of the likelihoo d to p erform inference
for a P oisson pro cess mo del, where the intensit y function dep ends on a Gaussian pro cess. The
resulting algorithms are one order of magnitude faster than state-of-the-art metho ds solving the
same problem. F urthermore, w e show that the same algorithms can be utilised for Ba y esian densit y
estimation, i.e. inferring a p osterior o v er densities for an observ ed set of p oin ts. Concluding the first
part, the inference problem for a Mark ov jump process mo del, namely the kinetic Ising mo del from
statistical ph ysics, is addressed using the new fav ourable representation of point process likelihoo ds.
The second part of the thesis is dev oted to the statistical description of a sp ecific instance of
p oin t pro cess data – the cell-resolv ed spiking activity of neurons. These data are b eliev ed to
reflect the information pro cessing in the brain, and are highly non–stationary . W e address the
problem of statistical mo delling suc h non–stationary spiking data. First, w e prop ose a con tin uous
time mo del accoun ting for effectiv e couplings and temp oral c hanges of the neuronal dynamics.
Deriving an efficien t inference algorithm, w e demonstrate that the mo del can capture activit y
structures of in–viv o recorded data, that are not related to an y controlled v ariables of the exp eriment.
Finally , w e prop ose a mo del whic h attempts to minimise the gap to the underlying system, based
on the assumption of observing a p opulation of in tegrate–and–fire neurons receiving common
non–stationary input. W e demonstrate ho w to efficiently ev aluate the mo del likelihoo d, suc h that
subsequen t inference can b e p erformed giv en spiking data recorded from a neuronal p opulation.
The no vel scalable inference algorithms for point process data, and the new description of non–
stationary spiking data presen ted in this thesis expand our abilit y to inv estigate large and complex
p oin t pro cess datasets and dra w meaningful conclusions from these data.
iii

iv

Zusammenfassung
Ankunftszeiten v on Flugzeugen, Positionen v on Autounfällen o der astronomisc hen Ob jekten im
W eltraum, Orte v on ökologisc hen Katastrophen, Impulse v on Neuronen usw. sind allesam t Daten,
die uns umgeb en und als Realisierung v on Punktprozessen b etrac htet w erden k önnen. Die Mo-
dellierung dieser Daten wird heutzutage immer wic htiger, w enn wir v ersuc hen, aus dieser ständig
w ac hsenden Datenmenge aussagekräftige Sc hlüsse zu ziehen. In der V ergangenheit wurden Mo delle
zur statistisc hen Beschreibung v on Punktprozessdaten v orgesc hlagen.
Die Extraktion der Mo dellparameter b ei der Beobac h tung v on Punktprozessdaten ist jedo c h
in der Regel eine Herausforderung. Die Punktprozessdic hte der beobach teten Daten gegeb en der
Mo dellparameter, auc h Lik eliho o d-F unktion genann t, ist in der Praxis sc h w er zu handhab en aufgrund
ihrer funktionalen F orm. Für eine Ba yes’sc he Inferenz, b ei der wir eine p osteriore V erteilung üb er
gew onnenen Mo dellparameter gegeb en der Daten anstreb en, ist die Aufgab e no c h anspruchsv oller.
Im ersten T eil dieser Arb eit k onzentrieren wir uns auf eine b estimm te Mo dellklasse für Punktpro-
zessdaten. Das zen trale Ob jekt v on Punktprozessmo dellen ist die nich tnegativ e In tensitätsfunktion,
die die W ahrsc heinlichk eit b estimm t, ein Ereignis an einer b eliebigen Stelle im Raum zu b eobac h-
ten. Um die Nic ht-Negativität zu gew ährleisten, wurden Punktprozessmo delle v orgesc hlagen, b ei
denen die In tensitätsfunktion nich t-linear v on den Mo dellparametern üb er eine sk alierte sigmoide
F unktion abhängt. Mit einer Mo dellaugmen tation durch laten te V ariablen zeigen wir, dass die
Punktprozessdic h te dieser Mo dellklasse in eine neuartige, v orteilhafte F orm gebrac h t w erden k ann,
die effizien te und schnelle Ba y es’sc he Inferenzalgorithmen ermöglic h t für eine praktisc h nutzbare
p osteriore V erteilung üb er die Mo dellparameter. Wir n utzen diese neue augmentierte Darstellung
der Lik eliho o d-F unktion, um Inferenz für ein P oisson-Prozessmo dell durc hzuführen, b ei dem die
In tensitätsfunktion v on einem Gaußprozess abhängt. Die resultierenden Algorithmen sind um eine
Größenordn ung sc hneller als mo derne Metho den, die das gleic he Problem lösen. Ansc hließend zeigen
wir, dass die gleic hen Algorithmen v erw endet w erden k önnen für eine Ba yes’sc he Dic h tesc hätzung,
d.h. die Inferenz einer p osterioren V erteilung üb er die Dic h te gegeb en eine b eobac h tete Menge von
Punkten. Absc hließend wird das Inferenzproblem für ein Mark o v-Sprung-Prozessmo dell, nämlic h
das kinetisc he Ising-Mo dell aus der statistisc hen Ph ysik, mit der neuen augmen tierten Darstellung
der Lik eliho o d-F unktion angegangen.
Der zw eite T eil der Arb eit widmet sich der statistisc hen Besc hreibung einer b estimm ten Instanz
v on Punktprozessdaten - der zellaufgelösten Spiking-Aktivität v on Neuronen, die im Allgemeinen
nic ht stationär ist. Wir befassen uns mit dem Problem der statistischen Modellierung solcher
nic ht stationärer Spiking-Daten. Zunäc hst sc hlagen wir ein k on tin uierlic hes Zeitmo dell v or, das
effektiv e neuronale K opplungen und zeitlic he V eränderung der Daten b erüc ksic h tigt. Mit Hilfe der
Herleitung eines effizien ten Inferenzalgorithmus zeigen wir, dass die inferierten Modellparameter
Aktivitätsstrukturen v on in–vivo aufgezeic hneten Daten aufzeigen können, die nic h t mit den
k on trollierten V ariablen des Exp erimen ts v erbunden sind. Sc hließlic h schlagen wir ein Modell vor, das
v ersuc h t, den Abstand zum zugrunde liegenden System zu minimieren, basierend auf der Annahme,
dass die P opulation von Inte gr ate-and-Fir e Neuronen b eobac h tet wird, die einen gemeinsamen nic h t-
stationären Input erfährt. Wir zeigen, wie man die Mo dell-Lik eliho o d-F unktion effizien t ev aluiert,
so dass mit Hilfe v on Spikingdaten, die von einer neuronalen P opulation aufgenommen wurden,
eine nac hfolgende Inferenz durchgeführt w erden k ann.
Die neuartigen sk alierbaren Inferenzalgorithmen für Punktprozessdaten und nic ht stationäre Spiking-
Daten erw eitern unsere Möglichk eiten , große und k omplexe Punktprozessdatensätze zu un tersuc hen
und aus diesen Daten neue und aussagekräftige Sc hlussfolgerungen zu ziehen.
v

Con ten ts
A c kno wledgemen ts i
Abstract (English/Deutsc h) iii
Con ten ts vii
1 In tro duction 1
I Efficien t Ba y esian inference for p oin t pro cesses 5
2 Gaussian represen tation of a p oin t pro cess lik eliho o d 7
3 Journal article: Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o-
c esses 11
4 Conference article: Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Den-
sity Mo del 47
5 Journal article: Inverse Ising pr oblem in c ontinuous time: A latent variable
appr o ach 59
6 Conjugacy b y augmen tation: A dditional mo dels & p oten tial extensions 69
I I Inference of mo dels for non-stationary spiking data 73
7 Statistical mo delling of spiking data: A brief in tro duction 75
8 Unpublished article: Bayesian network infer enc e fr om non-stationary spiking
data 77
9 Unpublished article: Inferring the c ol le ctive dynamics of neur onal p opulations
fr om single-trial spike tr ains using me chanistic mo dels 103
10 Conclusion 137
I I I App endix 139
A Augmen tation for GP m ulti–class classification . . . . . . . . . . . . . . . . . . . . . 141
B Alternativ e deriv ations of v ariational lo wer b ound . . . . . . . . . . . . . . . . . . . 144
Bibliograph y 147
Glossary 153
Con tributions 155
vii

Con ten ts
Cop yrigh t 157
viii

Chapter 1
In tro duction
In presen t da ys w e are surrounded b y an immeasurable amoun t of data. As w e try to make sense
of the constan t stream of data, one of the ma jor c hallenges w e are facing is finding meaningful
patterns and dra w conclusions from them. In the mac hine learning comm unit y dra wing conclusions
from data is often though t of as obtaining a mo del that explains statistical prop erties of the data
w ell. These mo dels should b e carefully designed, so that structures of in terest are revealed.
In the Ba y esian comm unit y ‘obtaining a mo del’ is defined as finding the p osterior densit y o v er the
mo del parameters
Z
giv en the observ ed data
D
. The p osterior is obtained b y Bayes’ rule (Stuart
and Ord 2010)
p ( Z |D ) = L ( D | Z ) p ( Z )
p ( D ) , (1)
where one assumes a lik eliho o d of data giv en a set of parameters
L
(
D | Z
) and also some prior b eliefs
for those parameters
p
(
Z
) . The denominator
p
(
D
) is the normalisation also kno wn as evidence, whic h
requires marginalising the nominator with resp ect to mo del parameters
Z
. Practical computation
of exp ectations with resp ect to the p osterior in Eq (1) is in general infeasible. Bayesian infer enc e
is concerned with finding a tractable p osterior form. A simple example of Ba y esian inference is
regression, where one assumes that data
D
are noisy measuremen ts of some underlying function,
whic h w e w an t to estimate. The t yp e of assumed noise dictates the lik eliho o d
L
(
D | Z
) . F urthermore,
one needs to assume whic h class of functions the target function b elongs to, b y defining the
parameters
Z
. Prior b eliefs (e.g. the function is con tinuous or differen tiable) determine the prior
p ( Z ) .
A sp ecific problem, where Eq
(1)
actually results in a tractable p osterior is Gaussian regres-
sion (Bishop 2006) . In this case the measuremen t noise is normally distributed, the underlying
function dep ends linearly on the parameters
Z
, and the prior
p
(
Z
) is a Gaussian densit y . Under
these assumptions the mo del is ‘conjugate’, i.e. the p osterior has the same form as the prior distri-
bution. Hence, Eq
(1)
yields a p osterior, for whic h w e are able to practically compute normalisation,
exp ectation, etc.
In general this is not the case, as other (non–Gaussian) noise mo dels and differen t types of
problems (e.g. classification) imp ose lik eliho o ds, for which posteriors are in tractable. F or these
cases, maximising the (logarithm of the) nominator with resp ect to the parameters
Z
is already
difficult and one is often required to utilise n umerical optimisation pro cedures. Those are often
slo w and contain themselv es parameters, that need to b e tuned to efficien tly arriv e at an optimal
solution. T o solve the Ba y esian problem one needs to obtain not only a p oin t estimate, but an
optimal p osterior distribution o v er the mo del parameters
Z
. This is usually in tractable, b ecause it
is practically infeasible to compute the normalisation constan t
p
(
D
) , and exp ectations with resp ect
1

Chapter 1. In tro duction
to the p osterior in Eq 1. T o circum v en t this issue man y approac hes resort to appro ximations, whic h
in turn can b e v ery slo w, such that they do not scale to large datasets
D
or require n umerical
optimisation, whic h come with the same problems mentioned abov e. T o obtain a tractable p osterior
man y Bay esian inference metho ds ha v e b een prop osed suc h as sampling sc hemes (Hastings 1970) ,
Laplace appro ximation (Bishop 2006) , exp ectation propagation (Mink a 2001; Opp er and Winther
2000) , v ariational metho ds (F eynman et al. 1964) , b elief propagation (Y edidia 2013) etc. The
optimisation problem b ecomes ev en more c hallenging for inference of sto c hastic pro cesses with an
infinite n umber of parameters Z .
In this thesis w e will address Bay esian inference for a sp ecific class of random pro cesses, where
the data are discrete ev ents in an observ ed space. Data of this t yp e app ear in a broad range
of applications, suc h as healthcare (Ahmed and Alkhamis 2009) , forestry (Pen ttinen and Sto y an
2000) , financial mark ets (Em brec hts et al. 2011) , w eather forecast (Kilsb y et al. 2007) , and crisis
prediction (Zammit-Mangion et al. 2012) . Given this t yp e of p oin t observ ations the main challenges
are (i) to infer the frequency of ev en ts at an y giv en p oin t in the observ ed space, and (ii) the
probabilit y of an even t b eing in a certain subspace.
The simplest scenario in whic h these questions can b e addressed assumes that all ev en ts are
indep enden t. Under this assumption the problem (i) is equiv alen t to estimating the in tensit y
function of a P oisson pro cess (Kingman 1993) . Problem (ii) is called densit y estimation (Silv erman
1986) and is closely related to problem (i) as w e will see in this thesis. Those fundamen tal problems
ha v e b een addressed in the Ba y esian comm unit y several times (A dams et al. 2009; Llo yd et al. 2014;
Murra y et al. 2009; Riihimäki and V ehtari 2014) , but model inferen ce suffers from the obstacles
stated ab o v e, i.e. they do not scale to problems with large datasets or require n umerical optimisation
sc hemes.
If the observ ed even ts are in terdep enden t, the mo dels men tioned ab o v e are not applicable an y
more. Hence another c hallenge, in addition to the aforemen tioned ones, is to create (iii) mo dels
that accoun t for dep endencies b et w een data p oin ts. A particular case of such in terdep enden t data
can b e encoun tered in neuroscience, where neuronal activit y from many neurons is recorded in
parallel. These data con tain time p oin ts and neuron iden tifiers of observ ed neuronal even ts, namely
action p oten tials also kno wn as spikes. This kind of data, also kno wn as spik e train data, will b e of
ma jor in terest in the latter part of this thesis. The sequence of spik es generated by eac h neuron is
b eliev ed to con tain the main information whic h is transmitted to other neurons (Riek e et al. 1997)
and allo ws neuronal ensembles to perform the computations required for p erception, b eha viour etc.
In general, past ev ents are imp ortan t for an accurate description of temp oral data and spiking data
are no exception to that (Da y an and Abb ott 2001) . In addition to temp oral dep endence, these
data are c haracterised by spatial dependencies, i.e. the in termingled activit y of cells the observed
neurons are synaptically connected to. Th us, for spiking data the c hallenge of quan tifying the
dep endence of ev en ts translates to the question of ho w to infer the effectiv e net w ork structure.
Sev eral mo dels ha v e b een suggested to tac kle this issue (Chornob o y et al. 1988; Pillo w et al. 2008;
Zeng et al. 2013) , and as b efore, inference often requires n umerical optimisation.
Solving the inference problem b ecomes ev en more complex, when the data are non–stationary , i.e.
the data statistics c hange ov er time, whic h is usually the case for spiking data. This p oses the
problem of (iv) ho w to infer a time–v arying mo del structure. Sev eral approac hes in neuroscience
addressed this issue with v arying set of assumptions (Cunningham and Y u 2014; Kim and Shinomoto
2012; P andarinath et al. 2018) .
The mo dels commonly c hosen for statistical descriptions of spiking data are purely phenomenological,
i.e. they provide a flexible statistical description without being physiologically constrained. There is
little w ork that (v) tries to use plausible mec hanistic mo dels, that are constrained b y incorp orating
a credible spiking mec hanism. Some w ork (Ladenbauer et al. 2018; Mullo wney and Iy engar 2008)
sho wed that lik eliho o ds can b e deriv ed and ev aluated efficien tly for a simple neuron mo del class.
These mo dels are broadly used to sim ulate plausible spiking data and are p opular b ecause they
2

allo w analytical in v estigation of neuronal net w orks, while preserving a minimal set of the neuron’s
bioph ysical prop erties (Gerstner and Kistler 2002) . How ev er, the attempts to p erform inference
with these mo dels giv en recorded spiking data are scarce, b ecause the ev aluation of lik eliho o ds is
though t to b e difficult.
Thesis outline
This thesis addresses sp ecific instances of problems (i)–(v), mainly in the context of Ba y esian
inference, and consists of t wo parts.
The thesis’ first part addresses the inference of mo dels that ha v e b een already prop osed in the
past (A dams et al. 2009; Glaub er 1963; Murra y et al. 2009) . W e derive no v el Ba y esian inference
sc hemes that are substantially more efficien t than those prop osed b efore. Our approac h relies
on iterativ e up dates, that are analytically tractable and do not require n umerical optimisation.
Sp ecifically , w e fo cus on the inference of 3 mo dels, that ha v e similar lik eliho o ds
L
(
D | Z
) and consider
the problems (i)–(iii), resp ectiv ely .
In c hapter 2, we discuss a particular class of lik eliho o ds, that is shared b y the mo dels men tioned
previously , and briefly sk etch ho w an alternativ e fa v ourable represen tation can b e obtained. With
sp ecific (Gaussian) priors the mo del b ecomes conditionally conjugate (Bishop and Tipping 2000; Blei
et al. 2017) , whic h allows for efficien t optimisation pro cedures where the updates hav e an analytical
form. Chapter 3 makes use of this represen tation for a doubly sto c hastic P oisson pro cess (A dams
et al. 2009) , where the in tensity to be inferred dep ends non–linearly on a Gaussian pro cess (GP).
W e sho w in chapter 4, that a GP densit y mo del (Murra y et al. 2009) can b e transformed to the
previous P oisson pro cess mo del, which allo ws us to use the same tec hniques for inference as in
c hapter 3. In c hapter 5, which constitutes the end of the first part, w e sho w that the describ ed
augmen tation is also applicable to a Marko v jump pro cess accoun ting for effectiv e couplings among
parallel binary pro cesses (Glaub er 1963; Zeng et al. 2013) .
The second part addresses questions (iv)–(v) in the con text of statistical mo delling of non–stationary
spiking data. Chapter 7 briefly introduces common problems in the statistical mo delling of these
data. In c hapter 8, we propose a non–stationary extension of the mo del discussed in c hapter 5, and
deriv e an efficien t Ba y esian inference algorithm. After v alidating the accuracy of the inferred results
for sim ulated data, we demonstrate practicalit y on exp erimen tally recorded spik e train data. Finally ,
in c hapter 9 w e tac kle the problem of inference for ph ysiologically constrain t mo dels for spiking
data. W e prop ose a neuronal mo del with a minimally plausible spiking mec hanism, namely the
leaky in tegrate–and–fire (LIF) neuron (Gerstner and Kistler 2002) . Unlik e the phenomenological
mo dels considered b efore, the mo del parameters ha v e a straightforw ard bioph ysical in terpretation.
Here, w e make use of the fact that the model likelihoo d can b e efficien tly ev aluated (Laden bauer
et al. 2018) . W e consider a non–stationary scenario, where observ ed neurons are driv en b y an
unkno wn doubly sto c hastic pro cess, whic h w e attempt to infer.
3

P art I
Efficien t Ba y esian inference for p oin t
pro cesses
5

Chapter 2
Gaussian represen tation of a p oin t
pro cess lik eliho o d
In the previous c hapter we men tioned, that exact Ba y esian inference is practically infeasible,
b ecause the p osterior in Eq
(1)
is in tractable. Ho wev er, in particular cases, suc h as linear regression,
discussed in c hapter 1, the p osterior is analytically tractable due to the mo del conjugacy . F or
non–conjugate mo dels c onditional conjugacy can b e ac hiev ed in particular cases by rewriting the
mo del lik eliho o d as exp ectation o ver a set of new laten t augmen tation v ariables (Meng and V an Dyk
1999) . Conditional conjugacy means that the mo del is conjugate for the original mo del parameters
giv en the augmentation v ariables and vice versa. Ba yesian inference for conditionally conjugate
mo dels can not b e solv ed exactly , but in general allo ws for more efficien t inference algorithms than
non–conjugate mo dels (Bishop and Tipping 2000; Blei et al. 2017; Meng and V an Dyk 1999) . In
the follo wing, we deriv e an augmen tation sc heme for a particul ar model class, which allo ws a new
conditionally conjugate represen tations for those mo dels.
The mo del class of in terest in this part has sp ecific prop erties. Observ ed data
D
=
{ x n } N
n =1
are
discrete ev ents in a con tin uous or discrete domain
x n ∈ X
, and
X
is observ ed completely . The
lik eliho o d for suc h p oint pr o c esses (Daley and V ere-Jones 2008) has the form
p ( D | Z ) = ∏
x n ∈D
Λ Z ( x n ) exp ( − ∫ X
Λ Z ( x ) d x ) , (2)
where Λ
Z
:
X → R +
is the ‘in tensit y’ or ‘rate’ function parametrised b y the mo del v ariables
Z
. The
pro duct on the righ t-hand side in Eq
(2)
corresp onds to the ev en ts
D
, while the in tegral accoun ts for
the lo cations where no ev en ts are observ ed. Eq
(2)
is also called c omplete data likeliho o d (Wilkinson
2006) . While not addressed here, Eq
(2)
can b e easily extended to situations where differen t t yp es
of ev ents exist. In the follo wing w e fo cus on mo dels that assume Gaussian priors
p
(
Z
) . Since a
Gaussian densit y is defined on the whole real space, but the in tensit y function is restricted to
p ositiv e real n um b ers, Λ
Z
(
·
) has to dep end non–linearly on
Z
. While man y choices of suc h link
functions are p ossible (exp onen tial, square, cum ulativ e Gaussian or an y mixture of non–negativ e
functions), in this w ork we focus on mo dels with a scaled sigmoidal link function
Λ Z ( x ) = c σ ( h Z ( x )) = c
1 + exp( − h Z ( x )) , (3)
where
c ∈ R +
and
h Z
:
X → R
is a linear function of mo del parameters
Z
. Mo dels of this form are
non–conjugate to a Gaussian densit y and hence inference with such priors is c hallenging.
Another complication arises, when the mo del parameters dep end on the space
X
. If e.g.
X ⊆ R d
,
then Eq
(2)
dep ends on a con tin uum of v ariables
Z
, due to the integral in the exponent. The
7

Chapter 2. Gaussian represen tation of a p oin t pro cess lik eliho o d
inference problem in Eq
(1)
yields a p osterior o v er an infinite dimensional ob ject, whic h is practically
infeasible. T o obtain a tractable p osterior o ver
Z
, w e ha v e to resort to appro ximations (Matthews
et al. 2016) , as w e will see in the subsequen t c hapters.
Appro ximate inference for complete data lik eliho o d
As w e hav e seen, the p osterior in Eq
(1)
is in tractable for mo dels previously discussed. Ho wev er,
a zo o of appro ximate inference metho ds for suc h problems exists. The ma jor b o dy of w ork relies
either on Mark ov Chain Mon te Carlo (MCMC) (Murra y et al. 2012) or v ariational inference
metho ds (Matthews et al. 2016) and this thesis is no exception to that. How ev er, the algorithms
w e prop ose div erge from previously prop osed approac hes.
MCMC Sampling algorithms for mo dels with complete data lik eliho o d are usually of the Metrop olis
Hastings t yp e, based on rejections (Adams et al. 2009; Murra y et al. 2012, 2009) . While long
Mark o v c hains con v erge theoretically to the correct p osterior, in practice, con v ergence is slo w and
the algorithms do not scale w ell with the amount of data.
On the other hand, v ariational algorithms assume that the appro ximate p osterior b elongs to a
certain family of densities, for whic h the v ariational problem is tractable. The goal is to minimise
the Kullbac k–Leibler div ergence b et w een the v ariational and the true p osterior. This is equiv alent to
maximising a lo w er b ound of the evidence
p
(
D
) (Bishop 2006; Blei et al. 2017) . Often one assumes
a v ariational p osterior densit y of a sp ecific functional form, e.g. a Gaussian density . The v ariational
lo w er b ound is then maximised with resp ect to the p osterior parameters via n umerical optimisation,
e.g. gradient ascen t algorithms (Hensman et al. 2015b; Llo yd et al. 2014) . These metho ds hav e b een
pro v en faster than sampling sc hemes and yield fairly go o d appro ximate p osteriors. The dra wbac k of
these approac hes is that the required gradien ts can b e solv ed analytically only for sp ecific functions
Λ
Z
(
x
) , sub ject to Gaussian priors with sp ecific k ernels and particular domains
X
, ev en under the
Gaussian p osterior assumption (Llo yd et al. 2014) . In other cases discretisation of the domain
X
,
sampling or n umerical appro ximations of the in v olv ed in tegrals are required (Flaxman et al. 2017;
Hensman et al. 2015b) .
In the follo wing sections w e sho w that for the mo del class with complete data lik eliho o d in Eq
(2)
,
sigmoid link function
(3)
, and a Gaussian prior, w e can deriv e an augmen ted represen tation of Eq
(2)
suc h that the mo del b ecomes conditionally conjugate. Conditioned on the new latent v ariables the
p osterior densit y o v er the mo del parameters
Z
is a Gaussian densit y , for whic h w e can analytically
deriv e the mean and cov ariance matrix. The analytical calculation of the Gaussian posterior is
the ma jor difference to the previously discussed approac hes, whic h require n umerical optimisation
pro cedures.
A c hieving conjugacy via v ariable augmen tation
As w e kno w from the case of Gaussian regression, the Gaussian lik eliho o d is conjugate to a Gaussian
prior. Hence, we aim at a Gaussian represen tation of Eq
(2)
with in tensity function of form as
in Eq
(3)
. First, w e fo cus on the pro duct of the in tensit y function at the observ ed ev en ts
D
, i.e.
∏ x n ∈D cσ ( h Z ( x n )) .
P óly a–Gamma augmen tation
P olson et al. (2013) show ed that the sigmoidal function can b e
rewritten as an infinite scale Gaussian mixture mo del
σ ( z ) = 1
2 ∫ R +
exp ( z
2 − z 2
2 ω ) p PG ( ω | 1 , 0) dω , (4)
8

where the precision
ω
of the Gaussian densit y is distributed according to a Póly a–Gamma densit y
p PG
(
·|
1
,
0) . This general P óly a–Gamma densit y
p PG
(
·| b 1 , b 2
) parametrised b y
b 1 , b 2
has sev eral
in teresting prop erties. It can b e sampled efficien tly , it is conjugate to
exp
(
− z
2 ω
) , and its momen ts
can b e computed analytically (P olson et al. 2013) . F or the pro duct o ver observ ations
D
in Eq
(2)
w e obtain a Gaussian representation in terms of laten t v ariables Z
∏
x n ∈D
c σ ( h Z ( x n )) ∝ c N ∏
x n ∈D ∫ R +
exp ( h Z ( x n )
2 − [ h Z ( x n )] 2
2 ω n ) p PG ( ω n | 1 , 0) dω n . (5)
This term is fully conditionally conjugate, i.e. Eq
(5)
is prop ortional to a P óly a–Gamma densit y
for eac h
ω n
, giv en the mo del parameters
Z
, and prop ortional to a Gaussian o v er
Z
giv en
{ ω n } N
n =1
.
F or mo dels with binomial lik eliho o ds the conditional conjugate lik eliho o d is ac hiev ed with this
augmen tation. It has b een utilised for Ba yesian logistic regression problems (Linderman et al. 2016;
W enzel et al. 2018) and other lik eliho o ds, that can b e written as pro ducts of sigmoids (Linderman
et al. 2015; Scott and Pillo w 2012) . In the follo wing w e aim at finding a similar represen tation for
the exp onen tial term in Eq (2).
Mark ed P oisson pro cess augmen tation
F or the sigmoidal link function the equalit y
σ
(
z
) =
1 − σ ( − z ) holds, and the exp onen t in Eq (2) can b e written as
exp ( − ∫ X
cσ ( h Z ( x )) d x ) = exp ( ∫ X
( σ ( − h Z ( x )) − 1) cd x ) . (6)
This equation has the form of a c haracteristic function of a Poisson process. Th us, Campb ell’s
theorem (Kingman 1993 , c hap. 3 ) allo ws us to rewrite Eq (6) as
exp ( ∫ X
( σ ( − h Z ( x )) − 1) Λ( x ) d x ) = E P Λ( x ) [ ∏
x ∈ Π X
σ ( − h Z ( x )) ] , with Λ( x ) = c, (7)
where the exp ectation is tak en with resp ect to the probabilit y measure
P Λ
(
x
) of a P oisson pro cess
with in tensity Λ(
x
) on domain
X
. Π
X
is a random set of p oin ts on this domain. In fact this
represen tation w as used to deriv e Metrop olis Hastings sampler for a Poisson process mo del (A dams
et al. 2009) , b eing part of the mo del class discussed here. Directly applying Eq
(7)
to Eq
(6)
do es
not yield the desired conjugate form of the lik eliho o d. Ho wev er, b y first applying the Póly a-Gamma
augmen tation (4), and then inv oking Eq (7), the conjugate form is ac hiev ed
exp ( ∫ X
( σ ( − h Z ( x )) − 1) c d x ) (8)
= exp ( ∫ X × R + ( 1
2 exp ( − h Z ( x )
2 − [ h Z ( x )] 2
2 ω ) − 1 ) p PG ( ω | 1 , 0) c dω d x )
= E P Λ( x ,ω ) ⎡
⎣ ∏
( x ,ω ) ∈ Π X × R +
1
2 exp ( − h Z ( x )
2 − [ h Z ( x )] 2
2 ω ) ⎤
⎦ ,
where the in tensity Λ(
x , ω
) =
c p PG
(
ω |
1
,
0) . In fact,
P Λ( x ,ω )
is a measure o v er a marke d Poisson
pr o c ess in the pro duct space
X × R +
, where the P ólya–Gamma v ariables are marks on the ev en ts
in the data domain
X
(Kingman 1993 , c hap. 5 ) . Note that a Poisson process measure is conjugate
to a P oisson pro cess lik eliho o d (Kingman 1993) .
F ull conditional conjugacy
By Eq
(5)
and Eq
(8)
w e achiev e a conditional conjugate repre-
sen tation of likelihoo d in Eq
(2)
. The resulting join t lik eliho o d of mo del v ariables
Z
and of the
laten t v ariables
{ ω n } N
n =1 ,
Π
X × R +
is conditionally conjugate for ev ery set of v ariables. This fact
allo ws deriving inference algorithms, whic h are m uc h more efficien t compared to others that can b e
9

Chapter 2. Gaussian represen tation of a p oin t pro cess lik eliho o d
obtained from w orking with Eq (2) directly (Meng and V an Dyk 1999) .
Inference algorithms for the augmen ted mo del
F or conditionally conjugate mo dels, blo c k Gibbs sampling (Geman and Geman 1984) is a MCMC
algorithm that is rejection free, and hence exp ected to con v erge m uc h faster than other Metrop olis–
Hastings algorithms discussed earlier (Meng and V an Dyk 1999) . F or the augmen ted mo del deriv ed
here, it is in fact p ossible to sample from eac h conditional p osterior efficien tly .
The new conjugate form of the mo del also allo ws for a v ariational mean–field algorithm (Bishop
2006; Blei et al. 2017) , where it is assumed that the mo del parameters
Z
are indep enden t of the
augmen tation v ariables
{ ω n } N
n =1 ,
Π
X × R +
. Due to the conjugacy , w e can derive the v ariational
mean–field p osterior analytically and no gradien t optimisation is required, unlik e the previously
discussed metho ds (Hensman et al. 2015b; Llo yd et al. 2014) .
F urthermore, the augmen tation allo ws to find the maxim um lik eliho o d or maxim um a p osteriori
estimate of
Z
exactly via an efficien t exp ectation–maximisation algorithm (EM) (Dempster et al.
1977; Meng and V an Dyk 1999) . T o obtain an appro ximate p osterior, one can p erform the Laplace
appro ximation (Bishop 2006) by calculating the Hessian of the original lik eliho o d
(2)
with resp ect
to the mo del v ariables Z .
Outline of part I
Ha ving established the common ground of the first part of this thesis, w e no w presen t a short
o v erview of the mo dels discussed in c hapter 3–5.
In c hapter 3 w e discuss the sigmoidal Gaussian Co x pro cess mo del, whic h was originally in tro duced
b y A dams et al. (2009) . In this w ork we deriv e the augmen tation sc heme outlined ab ov e in a more
rigorous w a y , together with deriving a v ariational mean field algorithm and an EM algorithm with
Laplace appro ximation. W e establish that our algorithms are one order of magnitude faster than
state-of-the-art inference metho ds for the same mo del, and are compatible with v ariational inference
algorithms for comp eting mo dels.
Chapter 4 deals with a Gaussian pro cess (GP) densit y mo del suggested b y Murray et al. (2009) .
This mo del is, strictly sp eaking, not part of the discussed mo del class, b ecause its lik eliho o d do es
not ha ve the form of Eq
(2)
. How ev er, w e sho w that with one additional v ariable augmentation
the required functional form of the lik eliho o d is obtained. While in c hapter 3 w e restrict ourselv es
to compact domains
X
, the GP density model allows to consider domains without b oundaries,
b y introducing base measures. W e dev elop the blo c k Gibbs sampler and a v ariational mean field
algorithm for appro ximate inference.
Finally , in c hapter 5 we apply the previously discussed augmen tation sc heme to a sp ecific Mark ov
jump pro cess mo del, namely the kinetic Ising mo del. It was prop osed first in statistical ph ysics to
describ e the dynamics of binary spins, whic h are coupled to each other (Glauber 1963) . In recen t
y ears the inverse problem (Nguyen et al. 2017) , i.e. obtaining the mo del parameters from observ ed
binary data, receiv ed increasing atten tion due to the use of suc h mo dels in neuroscience (Dunn
et al. 2015; Sc hneidman et al. 2006) . With the deriv ed augmen tation sc heme, w e establish the
EM algorithm for obtaining the
L
1 –p enalised maxim um lik eliho o d estimate exactly . W e mak e use
of the fact, that the Laplace prior can b e also rendered in to a Gaussian form, b y an additional
augmen tation (P on til et al. 2000) . W e dev elop a v ariational mean field algorithm for this mo del.
Chapter 6 discusses related w ork, limiting factors and future directions of research.
10

Chapter 3
Journal article: Efficient Bayesian
Infer enc e of Sigmoidal Gaussian
Cox Pr o c esses
Published in the journal Journal of Machine L e arning R ese ar ch (JMLR, Inc. and Microtome
Publishing, United States).
Authors:
Christian Donner 1 , 2 , Manfred Opp er 1 , 2
1 T echnisc he Univ ersität Berlin. 2 Bernstein Cen ter for Computational Neuroscience Berlin.
Details:
Submitted: D ecem b er 2017
A ccepted: Octob er 2018
URL: h ttp://jmlr.org/pap ers/v19/17-759.h tml
License: Creativ e Commons A ttribution (CC BY 4.0)
Chapter 3
This c hapter comprises the publication (Donner and Opp er 2018b) , which is authored
b y myself (CD), and Prof. Manfred Opp er (MO).
Con tributions :
CD and MO conceiv ed and designed the work. CD deriv ed the inference algorithms and dev elop ed
the Python co de. CD p erformed the n umerical exp erimen ts. CD wrote the man uscript with
substan tial contribution of MO.
Python co de on GitHub: https://gith ub.com/c hristiando/SGCP_Inference.git
11

Journal of Mac hine Learning Research 19 (2018) 1-34 Submitted 12/17; Revised 10/18; Published 11/18
Efficien t Ba y esian Inference of Sigmoidal Gaussian Co x
Pro cesses
Christian Donner [email protected]
Manfred Opp er [email protected]
A rtificial Intel ligenc e Gr oup
T e chnische Universit¨ at Berlin
Berlin, Germany
Editor: Ry an Adams
Abstract
W e presen t an appro ximate Ba y esian inference approach for estimating the in tensit y of an
inhomogeneous P oisson pro cess, where the intensit y function is mo delled using a Gaussian
pro cess (GP) prior via a sigmoid link function. Augmen ting the mo del using a latent
mark ed Poisson process and P´ oly a–Gamma random v ariables w e obtain a represen tation
of the lik eliho o d whic h is conjugate to the GP prior. W e estimate the p osterior using a
v ariational free–form mean field optimisation together with the framew ork of sparse GPs.
F urthermore, as alternativ e appro ximation we suggest a sparse Laplace’s method for the
p osterior, for whic h an efficien t exp ectation–maximisation algorithm is deriv ed to find the
p osterior’s mo de. Both algorithms compare w ell against exact inference obtained b y a
Mark ov Chain Mon te Carlo sampler and standard v ariational Gauss approac h solving the
same mo del, while b eing one order of magnitude faster. F urthermore, the p erformance and
sp eed of our metho d is comp etitiv e with that of another recen tly prop osed P oisson pro cess
mo del based on a quadratic link function, while not b eing limited to GPs with squared
exp onen tial k ernels and rectangular domains.
Keyw ords: P oisson pro cess; Cox process; Gaussian pro cess; data augmen tation; v aria-
tional inference
1. In tro duction
Estimating the in tensit y rate of discrete ev en ts ov er a con tin uous space is a common prob-
lem for real w orld applications suc h as mo deling seismic activit y (Ogata, 1998), neural data
(Brillinger, 1988), forestry (Sto y an and P en ttinen, 2000) and so forth. A particularly com-
mon approac h is a Ba y esian mo del based on a so–called Co x pro cess (Cox, 1955). The
observ ed ev en ts are assumed to b e generated from a P oisson pro cess, whose in tensit y func-
tion is mo deled as another random pro cess with a giv en prior probability measure. The
problem of inference for suc h t yp e of mo dels has also attracted in terest in the Ba y esian
mac hine learning comm unit y in recen t y ears. Møller et al. (1998); Brix and Diggle (2001);
Cunningham et al. (2008) assumed that the in tensit y function is sampled from a Gaussian
Pro cess (GP) prior (Rasm ussen and Williams, 2006). Ho w ever, to restrict the intensit y
function of the P oisson pro cess to nonnegativ e v alues, a common strategy is to c ho ose a
nonlinear link function whic h tak es the GP as its argumen t and returns a v alid in tensity .
Based on the success of v ariational appro ximations to deal with complex Gaussian pro cess
c
 2018 Christian Donner and Manfred Opp er.
License: CC-BY 4.0, see https://creativecommons . org/licenses/by/4 . 0/ . Attribution requiremen ts are pro vided
at http://jmlr . org/papers/v19/17- 759 . html .

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
12

Donner and Opper
mo dels, the inference problem for suc h Poisson models has attracted considerable in terest
in the mac hine learning comm unit y .
While p o w erful blac k–b o x v ariational Gaussian inference algorithms are a v ailable whic h
can b e applied to arbitrary link–functions, the choice of link –functions is not only crucial for
defining the prior o v er in tensities but can also b e imp ortan t for the effi ciency of v ariational
inference. The ’standard’ c hoice of Cox processes with an exp onential link function w as
treated in (Hensman et al., 2015). Ho w ev er, v ariational Gaussian inference for this link
function has the disadv an tage that the p osterior v ariance b ecomes decoupled from the
observ ations (Llo yd et al., 2015). 1 An in teresting c hoice is the quadratic link function of
(Llo yd et al., 2015) for whic h in tegrations o v er the data domain, whic h are necessary for
sparse GP inference, can b e (for sp ecific k ernel) computed analytically . 2 F or b oth mo dels,
the minimisation of the v ariational free energies is p erformed b y gradien t descent tec hniques.
In this pap er w e will deal with appro ximate inference for a mo del with a sigmoid link–
function. This mo del w as in tro duced b y (Adams et al., 2009) together with a MCMC
sampling algorithm whic h w as further impro v ed by (Gun ter et al., 2014) and (T eh and
Rao, 2011). Kiric henk o and v an Zan ten (2015) ha ve sho wn that the mo del has fa v ourable
(frequen tist) theoretical prop erties pro vided priors and h yp erparameters are c hosen ap-
propriately . In con trast to a direct v ariational Gaussian appro ximation for the p osterior
distribution of the laten t function, we will in tro duce an alternativ e t yp e of v ariational ap-
pro ximation whic h is sp ecially designed for the sigmoidal Gaussian Cox pr o c ess . W e build
on recen t w ork on Ba y esian logistic regression by data augmen tation with P´ oly a–Gamma
random v ariables (P olson et al., 2013). This approach w as already used in com bination
with GPs (Linderman et al., 2015; W enzel et al., 2017), for sto c hastic pro cesses in discrete
time (Linderman et al., 2017), and for jump pro cesses (Donner and Opp er, 2017). W e ex-
tend this metho d to an augmen tation b y a laten t, mark ed P oisson pro cess, where the marks
are distributed according to a P´ oly a–Gamma distribution. 3 In this w a y , the augmen ted
lik eliho o d b ecomes conjugate to a GP distribution. Using a com bination of a mean–field
v ariational appro ximation together with sparse GP appro ximations (Csat´ o and Opper, 2002;
Csat´ o, 2002; Titsias, 2009) w e obtain explicit analytical v ariational up dates leading to fast
inference. In addition, w e sho w that the same augmen tation can b e used for the computa-
tion of the maxim um a p osteriori (MAP) estimate b y an exp ectation–maximisation (EM)
algorithm. With this w e obtain a Laplace appro ximation to the non–augmen ted p osterior.
The pap er is organised as follo ws: In section 2, w e in tro duce the sigmoidal Gaussian
Co x pro cess mo del and its transformation b y the v ariable augmen tation. In section 3, w e
deriv e a v ariational mean field metho d and an EM–algorithm to obtain the MAP estimate,
follo w ed b y the Laplace appro ximation of the p osterior. Both metho ds are based on a
sparse GP appro ximation to mak e the infinite dimensional problem tractable. In section 4,
w e demonstrate the p erformance of our metho d on syn thetic datasets and compare with
the results of a Mon te Carlo sampling metho d for the mo del and the v ariational appro xi-
mation of Hensman et al. (2015), whic h w e mo dify to solv e the Co x–pro cess mo del with the
scaled sigmoid link function. Then w e compare our metho d to the state-of-the-art inference
1. Samo and Rob erts (2015) prop ose an efficien t approximate sampling sc heme.
2. F or a frequen tist nonparametric approach to this model, see (Flaxman et al., 2017). F or a Ba y esian
extension see (W alder and Bishop, 2017).
3. F or a differen t application of marked P oisson pro cesses, see (Llo yd et al., 2016).
2

13

Sigmoid al Ga ussian Cox Pr ocess Inference
algorithm (Llo yd et al., 2015) on artificial and real datasets with up to 10 4 observ ations.
Section 5 presen ts a discussion and an outlo ok.
2. The Inference problem
W e assume that N ev ents D = { x n } N
n =1 are generated b y a P oisson pro cess. Eac h p oin t
x n is a d –dimensional v ector in the compact domain X ⊂ R d . The goal is to infer the
v arying intensity function Λ( x ) (the mean measure of the pro cess) for all x ∈ X based on
the lik eliho o d
L ( D | Λ) = exp  − Z X
Λ( x ) d x  N
Y
n =1
Λ( x n ) ,
whic h is equal (up to a constan t) to the densit y of a P oisson pro cess ha ving in tensit y Λ (see
App endix C and (Konstan top oulos et al., 2011)) with resp ect to a P oisson pro cess with unit
in tensit y . In a Ba y esian framew ork, a prior o ver the in tensit y mak es Λ a random pro cess.
Suc h a doubly sto c hastic p oint process is called Cox pr o c ess (Co x, 1955). Since one needs
Λ( x ) ≥ 0, Adams et al. (2009) suggested a reparametrization of the intensit y function b y
Λ( x ) = λσ ( g ( x )), where σ ( x ) = (1 + e − x ) − 1 is the sigmoid function and λ is the maximum
in tensit y rate. Hence, the in tensity Λ( x ) is p ositiv e ev erywhere, for an y arbitrary function
g ( x ) : X → R and the inference problem is to determine this function. Throughout this
w ork w e assume that g ( · ) will b e mo delled as a GP (Rasm ussen and Williams, 2006) and
the resulting pro cess is called sigmoidal Gaussian Cox pr o c ess . The likelihoo d for g b ecomes
L ( D | g , λ ) = exp  − Z X
λσ ( g ( x )) d x  N
Y
n =1
λσ ( g n ) , (1)
where g n .
= g ( x n ). F or Ba y esian inference w e define a GP prior measure P GP with zero
mean and co v ariance k ernel k ( x , x 0 ) : X × X → R + . λ has as prior densit y (with resp ect to
the ordinary Leb esgue measure) p ( λ ) which w e tak e to b e a Gamma densit y with shap e-,
and rate parameter α 0 and β 0 , resp ectiv ely . Hence, for the prior w e get the pro duct measure
dP prior = dP GP × p ( λ ) dλ . The p osterior densit y p (with resp ect to the prior measure) is
giv en b y
p ( g , λ |D ) .
= dP p osterior
dP prior
( g , λ |D ) = L ( D | g , λ )
E P prior [ L ( D | g , λ )] . (2)
The normalising exp ectation in the denominator on the righ t hand side is with resp ect to
the probabilit y measure P prior . T o deal with the infinite dimensionalit y of GPs and P oisson
pro cesses w e require a minim um of extra notation. W e in tro duce densities or R adon–
Niko d´ ym derivatives suc h as defined in Equation (2) (see App endix C or de G. Matthews
et al. (2016)) with resp ect to infinite dimensional measures b y b oldface sym b ols p ( z ). On
the other hand, non–b old densities p ( z ) denote densities in the ‘classical’ sense, whic h means
they are with resp ect to Leb esgue measure d z .
Ba y esian inference for this mo del is kno wn to b e doubly in tractable (Murra y et al., 2006).
The lik eliho o d in Equation (1) con tains the in tegral of g o v er the space X in the exp onen t
and the normalisation of the p osterior in Equation (2) requires calculating exp ectation of
Equation (1). In addition inference is hamp ered b y the fact, that likelihoo d (1) dep ends
3

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
14

Donner and Opper
non–linearly on g (through sigmoid and exp onen t of sigmoid). In the follo wing w e tac kle
this b y an augmen tation sc heme for the lik eliho o d, suc h that it b ecomes conjugate to a GP
prior and w e subsequen tly can deriv e an analytic form of a v ariational p osterior giv en one
simple mean field assumption (Section 3).
2.1 Data augmen tation I: Laten t P oisson pro cess
W e will briefly in tro duce a data augmen tation sc heme b y a latent P oisson pro cess whic h
forms the basis of the sampling algorithm of Adams et al. (2009). W e will then extend
this metho d further to an augmen tation b y a marke d Poisson process. W e fo cus on the
exp onen tial term in Equation (1). Utilizing the w ell kno wn prop erty of the sigmoid that
σ ( x ) = 1 − σ ( − x ) w e can write
exp  − Z X
λσ ( g ( x )) d x  = exp  − Z X
(1 − σ ( − g ( x ))) λd x  . (3)
The left hand side has the form of a c haracteristic functional of a P oisson pro cess. Generally ,
for a random set of p oin ts Π Z = { z m ; z m ∈ Z } on a space Z and with a function h ( z ), this
is defined as
E P Λ 
 Y
z m ∈ Π Z
e h ( z m ) 
 = exp  − Z Z  1 − e h ( z )  Λ( z ) d z  , (4)
where P Λ is the probabilit y measure of a P oisson pro cess with in tensit y Λ( z ). Equation (4)
can b e deriv ed b y Campb ell’s theorem (see App endix A and (Kingman, 1993, c hap. 3))
and iden tifies a P oisson pro cess uniquely .
Setting h ( z ) = ln σ ( − g ( z )), and Z = X , and com bining Equation (3) and (4) w e obtain
the lik eliho o d used b y Adams et al. (2009, Eq. 4). Ho w ev er, in this w ork w e mak e use of
another augmen tation, b efore inv oking Campb ell’s theorem. This will result in a lik eliho o d
whic h is conjugate to the mo del priors and further simplifies inference.
2.2 Data augmen tation I I: P´ oly a–Gamma v ariables and mark ed P oisson
pro cess
F ollo wing P olson et al. (2013) w e represent the in v erse of the h yp erb olic cosine as a scaled
Gaussian mixture mo del
cosh − b ( z / 2) = Z ∞
0
e − z 2
2 ω p PG ( ω | b, 0) dω , (5)
where p PG is a P´ olya–Gamma densit y (App endix B). W e further define the tilte d P´ oly a–
Gamma densit y b y
p PG ( ω | b, c ) ∝ e − c 2
2 ω p PG ( ω | b, 0) , (6)
where b > 0 and c are parameters. W e will not need an explicit form of this densit y , since
the subsequen tly deriv ed inference algorithms will only require the first momen ts. Those
can b e obtained directly from the momen t generating function, whic h can b e calculated
straigh tforw ardly from Equation (5) and (6) (see App endix B). Equation (5) allo ws us to
4

15

Sigmoid al Ga ussian Cox Pr ocess Inference
rewrite the sigmoid function as
σ ( z ) = e z
2
2 cosh( z
2 ) = Z ∞
0
e f ( ω ,z ) p PG ( ω | 1 , 0) dω , (7)
where w e define
f ( ω , z ) .
= z
2 − z 2
2 ω − ln 2 .
Setting z = − g ( x ) in Equation (3) and substituting Equation (7) w e get
exp  − Z X
λ (1 − σ ( − g ( x ))) d x  = exp  − Z X × R +  1 − e f ( ω , − g ( x ))  p PG ( ω | 1 , 0) λdω d x  .
(8)
Finally , w e apply Campb ell’s theorem (Equation (4)) to Equation (8). The space is a
pro duct space Z = ˆ
X .
= X × R + and the intensit y Λ( x , ω ) = λp PG ( ω | 1 , 0). This results in
the final represen tation of the exp onen tial in Equation (8)
exp  − Z ˆ
X  1 − e f ( ω , − g ( x ))  Λ( x , ω ) dω d x  = E P Λ 
 Y
( x ,ω ) m ∈ Π ˆ
X
e f ( ω m , − g m ) 
 .
In terestingly , the new P oisson pro cess Π ˆ
X with measure P Λ has the form of a marke d P oisson
pro cess (Kingman, 1993, c hap. 5), where the laten t P´ olya-Gamma v ariables ω m denote the
‘marks’ b eing indep enden t random v ariables at eac h lo cation x m . It is straigh tforward to
sample suc h pro cesses b y first sampling the inhomogeneous P oisson pro cess on domain X
(for example b y ‘thinning’ a pro cess with constan t rate (Lewis and Shedler, 1979; Adams
et al., 2009)) and then dra wing a mark ω on eac h ev en t indep endently from the densit y
p PG ( ω | 1 , 0).
Finally , using the P´ oly a–Gamma augmentation also for the discrete lik eliho o d factors
corresp onding to the observ ed ev en ts in Equation (1) w e obtain the following join t lik eliho o d
of the mo del
L ( D , ω N , Π ˆ
X | g , λ ) .
= dP join t
dP aug
( D , ω N , Π ˆ
X | g , λ )
= Y
( x ,ω ) m ∈ Π ˆ
X
e f ( ω m , − g m )
N
Y
n =1
λe f ( ω n ,g n ) ,
(9)
where w e define the prior measure of augmen ted v ariables as P aug = P Λ × P ω N and where
ω N = { ω n } N
n =1 are the P´ oly a–Gamma v ariables for the observ ations D with the prior
measure dP ω N = Q N
n =1 p ( ω n | 1 , 0) dω n . This augmen ted represen tation of the lik eliho o d
con tains the function g ( · ) only linearly and quadratically in the exp onen ts and is thus
conjugate to the GP prior of g ( · ). Note that the original lik eliho o d in Equation (1) can b e
reco v ered b y E P aug  L ( D , ω N , Π ˆ
X | g , λ )  = L ( D | g , λ ).
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
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Donner and Opper
3. Inference in the augmen ted space
Based on the augmen tation w e define a p osterior densit y for the join t mo del with resp ect
to the pro duct measure P prior × P aug
p ( ω N , Π ˆ
X , g , λ |D ) .
= dP p osterior
d ( P prior × P aug ) ( ω N , Π ˆ
X , g , λ |D )
= L ( D , ω N , Π ˆ
X | g , λ )
L ( D ) ,
(10)
where the denominator is the marginal lik eliho o d L ( D ) = E P prior × P aug  L ( D , ω N , Π ˆ
X | g , λ )  .
The p osterior densit y of Equation (10) could b e sampled using Gibbs sampling with explicit,
tractable conditional densities. Similar to the v ariational appro ximation in the next section,
one can sho w that the conditional measure of the p oin t sets Π ˆ
X and the v ariables ω N , given
the function g ( · ) and maximal in tensity λ is a pro duct of a sp ecific mark ed P oisson pro cess
and indep enden t (tilted) P´ oly a–Gamma densities. On the other hand, the distribution o v er
function g ( · ) conditioned on Π ˆ
X and ω N is a Gaussian pro cess. Note, ho w ever, one needs
to sample this GP only at the finite p oin ts x m in the random set Π ˆ
X and the fixed set D .
3.1 V ariational mean–field appro ximation
F or v ariational inference one assumes that the desired p osterior probabilit y measure b elongs
to a family of measures for whic h the inference problem is tractable. Here w e mak e a
simple structured mean field assumption in order to fully utilise its conjugate structure:
W e appro ximate the p osterior measure b y
P p osterior ( ω N , Π ˆ
X , g , λ |D ) ≈ Q 1 ( ω N , Π ˆ
X ) × Q 2 ( g , λ ) , (11)
meaning that the dep endencies b et w een the P´ oly a–Gam ma v ariables ω N and the marked
P oisson pro cess Π ˆ
X on the one hand, and the function g and the maximal in tensit y λ on the
other hand, are neglected. As we will see in the follo wing, this simple mean–field assumption
allo ws us to deriv e the p osterior appro ximation analytically .
The v ariational appro ximation is optimised b y minimising the Kullbac k–Leibler div er-
gence b et w een exact and appro ximated p osteriors. This is equiv alen t to maximising the
lo w er b ound on the marginal lik eliho o d of the observ ations
L ( q ) = E Q  log  L ( D , ω N , Π ˆ
X | g , λ )
q 1 ( ω N , Π ˆ
X ) q 2 ( g , λ )  ≤ log L ( D ) , (12)
where Q is the probabilit y measure of the v ariational p osterior in Equation (11) and w e
in tro duced appro ximate lik eliho o ds
q 1 ( ω N , Π ˆ
X ) .
= dQ 1
dP aug
( ω N , Π ˆ
X ) , q 2 ( g , λ ) .
= dQ 2
dP prior
( g , λ ) .
Using standard argumen ts for mean field v ariational inference (Bishop, 2006, chap. 10)
and Equation (11), one can then sho w that the optimal factors satisfy
ln q 1  ω N , Π ˆ
X  = E Q 2  log L ( D , ω N , Π ˆ
X | g , λ )  + const. (13)
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17

Sigmoid al Ga ussian Cox Pr ocess Inference
and
ln q 2 ( g , λ ) = E Q 1  log L ( D , ω N , Π ˆ
X | g , λ )  + const. , (14)
resp ectiv ely . These results lead to an iterativ e sc heme for optimising q 1 and q 2 in order
to increase the lo w er b ound in Equation (12) in ev ery step. F rom the structure of the
lik eliho o d one deriv es t w o further factorisations:
q 1 ( ω N , Π ˆ
X ) = q 1 ( ω N ) q 1 (Π ˆ
X ) , (15)
q 2 ( g , λ ) = q 2 ( g ) q 2 ( λ ) , (16)
where the densities are defined with resp ect to the measures dP ( ω N ) , dP Λ , dP GP , and
p ( λ ) dλ , resp ectiv ely . The subsequen t section describ es these up dates explicitly .
Optimal P´ oly a–Gamma densit y F ollo wing Equation (13) and (15) w e obtain
q 1 ( ω N ) =
N
Y
n =1
exp  − c ( n )
1
2 ω n 
cosh − 1  c ( n )
1 / 2  =
N
Y
n =1
p PG  ω n | 1 , c ( n )
1 
p PG ( ω n | 1 , 0) ,
where the factors are tilts of the prior P´ oly a-Gamma densities (see Equation (6) and Ap-
p endix B) with c ( n )
1 = p E Q 2 [ g 2
n ]. By simple densit y transformation we obtain the densit y
with resp ect to the Leb esgue measure as
q 1 ( ω N ) = q 1 ( ω N )    
dP ω N
d ω N    
=
N
Y
n =1
p PG  ω n | 1 , c ( n )
1  , (17)
b eing a pro duct of tilte d P´ oly a–Gamma densities.
Optimal P oisson pro cess Using Equation (13) and (15) w e obtain
q 1 (Π ˆ
X ) = Q ( x ,ω ) m ∈ Π ˆ
X e E Q 2 [ f ( ω m , − g m )] λ 1
exp  R ˆ
X  e E Q 2 [ f ( ω , − g ( x ))] − 1  λ 1 p PG ( ω | 1 , 0) d x dω  , (18)
with λ 1 .
= e E Q 2 [log λ ∗ ] . Note, that E Q 2 [ f ( ω m , − g m )] in v olv es the exp ectations E Q 2 [ g m ] and
E Q 2  ( g m ) 2  . One can sho w, that Equation (18) is again a marked P oisson pro cess with
in tensit y
Λ 1 ( x , ω ) = λ 1
exp  − E Q 2 [ g ( x )]
2 
2 cosh  c 1 ( x )
2  p PG ( ω | 1 , c 1 ( x ))
= λ 1 σ ( − c 1 ( x )) exp  c 1 ( x ) − E Q 2 [ g ( x )]
2  p PG ( ω | 1 , c 1 ( x ))
(19)
where c 1 ( x ) = p E Q 2 [ g ( x ) 2 ] (for a pro of see App endix D).
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
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Donner and Opper
Optimal Gaussian pro cess F rom Equation (14) and (16) w e obtain the optimal ap-
pro ximation of the p osterior lik eliho o d (note that this is defined relativ e to GP prior)
q 2 ( g ) ∝ e U ( g ) ,
where the effectiv e log–lik eliho o d is giv en b y
U ( g ) = E Q 1 
 X
( x ,ω ) m ∈ Π ˆ
X
f ( ω m , − g m ) 
 +
N
X
n =1
E Q 1 [ f ( ω n , g ( x n ))] .
The first exp ectation is o v er the v ariational P oisson pro cess Π ˆ
X and the second one o v er
the P´ oly a–Gamma v ariables ω N . These can b e easily ev aluated (see App endix A) and one
finds
U ( g ) = − 1
2 Z X
A ( x ) g ( x ) 2 d x + Z X
B ( x ) g ( x ) d x , (20)
with
A ( x ) =
N
X
n =1
E Q 1 [ ω n ] δ ( x − x n ) + Z ∞
0
ω Λ 1 ( x , ω ) dω ,
B ( x ) = 1
2
N
X
n =1
δ ( x − x n ) − 1
2 Z ∞
0
Λ 1 ( x , ω ) dω ,
where δ ( · ) is the Dirac delta function. The exp ectations and in tegrals o v er ω are
E Q 1 [ ω n ] = 1
2 c ( n )
1
tanh c ( n )
1
2 ! ,
Z ∞
0
Λ 1 ( x , ω ) dω = λ 1 σ ( − c 1 ( x )) exp  c 1 ( x ) − E Q 2 [ g ( x )]
2  .
= Λ 1 ( x ) ,
Z ∞
0
ω Λ 1 ( x , ω ) dω = 1
2 c 1 ( x ) tanh  c 1 ( x )
2  Λ 1 ( x ) .
The resulting v ariational distribution defines a Gaussian pro cess. Because of the mean–
field assumption the in tegrals in Equation (20) do not require in tegration o v er random
v ariables, but only solving t w o deterministic in tegrals ov er space X . How ev er, those in tegrals
dep end on function g o v er the en tire space and it is not p ossible for a general kernel to
compute the marginal p osterior densit y at an input x in closed form. F or sp ecific GP
k ernel op erators, whic h are the in v erses of differential operators, a solution in terms of
linear partial differen tial equations w ould b e p ossible. This could b e of sp ecial in terest for
one–dimensional problems where Matern k ernels with in teger parameters (Rasm ussen and
Williams, 2006) fulfill this condition. Here, the problem b ecomes equiv alent to inference
for a (con tin uous time) Gaussian hidden Mark o v mo del and could b e solv ed b y p erforming
a forw ard–bac kw ard algorithm (Solin, 2016). This w ould reduce the computations to the
solution of ordinary differen tial equations. W e will discuss details of suc h an approach
elsewhere. T o deal with general kernels w e will resort instead to a the w ell kno wn v ariational
sparse GP appro ximation with inducing p oin ts.
8

19

Sigmoid al Ga ussian Cox Pr ocess Inference
Optimal sparse Gaussian pro cess The sparse v ariational Gaussian approximation fol-
lo ws the standard approac h (Csat´ o and Opp er, 2002; Csat´ o, 2002; Titsias, 2009) and its
generalisation to a con tin uum lik eliho o d (Batz et al., 2018; de G. Matthews et al., 2016).
F or completeness, we repeat the deriv ation here and more detailed in App endix E. W e
appro ximate q 2 ( g ) b y a sparse lik eliho o d GP q s
2 ( g ) with resp ect to the GP prior
dQ s
2
dP ( g ) = q s
2 ( g s ) , (21)
whic h dep ends only on a finite dimensional v ector of function v alues g s = ( g ( x 1 ) , . . . , g ( x L )) >
at a set of inducing p oints { x l } L
l =1 . With this approac h it is again p ossible to marginalise
out exactly all the infinitely man y function v alues outside of the set of inducing p oin ts. The
sparse lik eliho o d q s
2 is optimised b y minimising the Kullbac k–Leibler div ergence
D KL ( Q s
2 k Q 2 ) = E Q s
2  log q s
2 ( g )
q 2 ( g )  .
A short computation (App endix E) sho ws that
q s
2 ( g s ) ∝ e U s ( g s ) with U s ( g s ) = E P ( g | g s ) [ U ( g )] ,
where the conditional exp ectation is with resp ect to the GP prior measure giv en the function
g s at the inducing p oin ts. The explicit calculation requires the conditional exp ectations of
g ( x ) and of ( g ( x )) 2 . W e get
E P ( g | g s ) [ g ( x )] = k s ( x ) > K − 1
s g s , (22)
where k s ( x ) = ( k ( x , x 1 ) , . . . , k ( x , x L )) > and K s is the k ernel matrix b et w een inducing
p oin ts. F or the second exp ectation, w e get
E P ( g | g s )  g 2 ( x )  =  E P ( g | g s ) [ g ( x )]  2 + const. (23)
The constan t equals the conditional v ariance of g ( x ) whic h do es not dep end on the sparse
set g s , but only on the lo cations of the sparse p oin ts. Because w e are dealing no w with
a finite problem w e can define the ‘ordinary’ p osterior densit y of the GP at the inducing
p oin ts with resp ect to the Leb esgue measure d g s . F rom Equation (20), (22), and (23),
w e conclude that the sparse p osterior at the inducing v ariables is a m ultiv ariate Gaussian
densit y
q s
2 ( g s ) = N ( µ s
2 , Σ s
2 ) , (24)
with the co v ariance matrix given b y
Σ s
2 =  K − 1
s Z X
A ( x ) k s ( x ) k s ( x ) > d x K − 1
s + K − 1
s  − 1
, (25)
and the mean
µ s
2 = Σ s
2  K − 1
s Z X
B ( x ) k s ( x ) d x  . (26)
In con trast to other v ariational approximations (see for example (Llo yd et al., 2015; Hens-
man et al., 2015)) w e obtain a closed analytic form of the v ariational p osterior mean and
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
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Donner and Opper
co v ariance which holds f or arbitrary GP k ernels. Ho w ever, these results dep end on finite
dimensional in tegrals o v er the space X which cannot be computed analytically . This is dif-
feren t to the sparse appro ximation for the P oisson mo del with square link function (Llo yd
et al., 2015), where similar in tegrals in the case of the squared exp onen tial k ernel can b e ob-
tained analytically . Hence, w e resort to a simple Mon te–Carlo in tegration, where inte gr ation
p oints are sampled uniformly on X as
I F = Z X
F ( x ) d x ≈ |X |
R
R
X
r =1
F ( x r ) .
The set of in tegration p oin ts { x r } R
r =1 is dra wn uniformly from the space X .
Finally , from Equation (21) and (24) w e obtain the mean function and the v ariance of
the sparse appro ximation for ev ery p oin t x ∈ X , which is
µ 2 ( x ) = E Q 2 [ g ( x )] = k s ( x ) > K − 1
s µ s
2 , (27)
and v ariance
( s 2 ( x )) 2 = k ( x , x ) − k s ( x ) > K − 1
s  I − Σ s
2 K − 1
s  k s ( x ) , (28)
where I is the iden tit y matrix.
Optimal densit y for maximal in tensit y λ F rom Equation (14) w e iden tify the optimal
densit y as a Gamma densit y
q 2 ( λ ) = Gamma( λ | α 2 , β 2 ) = β α 2
2 ( λ ) α 2 − 1 e − β 2 λ
Γ( α 2 ) , (29)
where α 2 = N + E Q 1 [ 1 Π ( x )] + α 0 , β 2 = β 0 + R X d x and Γ( · ) is the gamma function. 1 Π ( x )
denotes the indicator function b eing 1 if x ∈ Π and 0 otherwise and the integral is again
solv ed b y Mon te Carlo in tegration. This defines the required exp ectations for up dating q 1
b y E Q 2 [ λ ] = α 2
β 2 and E Q 2 [log λ ] = ψ ( α 2 ) − log β 2 , where ψ ( · ) is the digamma function.
Hyp erparameters Hyp erparameters of the mo del are (i) the co v ariance parameters θ
of the GP , (ii) the lo cations of the inducing p oin ts { x l } L
l =1 , and (iii) the prior parameters
α 0 , β 0 for the maximal in tensity λ . The co v ariance parameters (i) θ are optimised b y
gradien t ascen t follo wing the gradien t of the low er b ound in Equation (12) with resp ect to
θ (App endix F). As gradien t ascen t algorithm w e emplo y the AD AM algorithm (Kingma
and Ba, 2014). W e p erform alw a ys one step after the v ariational p osterior q is up dated as
describ ed b efore. (ii) The lo cations of the sparse GP { x l } L
l =1 could in principle b e optimised
as w ell, but w e k eep them fixed and p osition them on a regular grid o v er the space X .
F rom this c hoice it follo ws that K s is a T o eplitz matrix, when the kernel is translationally
in v ariant. This could b e inv erted in O ( L (log L ) 2 ) instead of O ( L 3 ) op erations (Press et al.,
2007) but w e do not emplo y this fact. Finally , (iii) the v alue for prior parameters α 0 and β 0
are c hosen suc h that p ( λ ) has a mean t wice and standard deviation once the intensit y one
w ould exp ect for a homogeneous P oisson Pro cess observing D . The complete v ariational
pro cedure is outlined in Algorithm 1.
10

21

Sigmoid al Ga ussian Cox Pr ocess Inference
Algorithm 1: V ariational Ba y es algorithm for sigmoidal Gaussian Co x pro cess.
Init: E Q [ g ( x )] , E Q  ( g ( x )) 2  at D and in tegration p oin ts, and E Q [ λ ] , E Q [log λ ]
1 while L not c onver ge d do
2 Up date q 1
3 PG distributions at observ ations : q 1 ( ω N ) with Eq. (17)
4 Rate of laten t pro cess : Λ 1 ( x , ω ) at in tegration p oin ts with Eq. (19)
5 Up date q 2
6 Sparse GP distribution : Σ s
2 , µ s
2 with Eq. (25), (26)
7 GP at D and in tegration p oin ts : E Q 2 [ g ( x )] , E Q 2  ( g ( x )) 2  with
Eq. (27), (28)
8 Gamma-distribution of λ : α 2 , β 2 with Eq. (29)
9 Up date k ernel parameters with gradien t ascen t
10 end
3.2 Laplace appro ximation
In this section w e will sho w that our v ariable augmen tation metho d is w ell suited for com-
puting a Laplace appro ximation (Bishop, 2006, c hap. 4) to the joint posterior of the GP
function g ( · ) and the maximal in tensit y λ as an alternativ e to the previous v ariational
sc heme. T o do so w e need the maxim um a p osteriori (MAP) estimate (equal to the mo de
of the p osterior distribution) and a second order T a ylor expansion around this mo de. The
augmen tation metho d will b e used to compute the MAP estimator iterativ ely using an EM
algorithm.
Obtaining the MAP estimate In general, a prop er definition of the p osterior mo de
w ould b e necessary , b ecause the GP p osterior is o v er a space of functions, whic h is an
infinite dimensional ob ject and do es not ha v e a density with respect to Leb esgue measure.
A p ossibilit y to a v oid this problem w ould b e to discretise the spatial in tegral in the lik eliho o d
and to appro ximate the p osterior b y a multiv ariate Gaussian densit y for whic h the mo de
can then b e computed b y setting the gradien t equal to zero. In this pap er, w e will use a
differen t approac h whic h defines the mo de directly in function space and allo ws us to utilise
the sparse GP appro ximation dev elop ed previously for the computations. A mathematically
prop er w a y w ould b e to deriv e the MAP estimator b y maximising a prop erly p enalised log–
lik eliho o d. As discussed e.g. in Rasm ussen and Williams (2006, c hap. 6) for GP mo dels
with lik eliho o ds whic h dep end on finitely many inputs on ly , this p enalt y is giv en b y the
squared repro ducing k ernel Hilb ert space (RKHS) norm that corresp onds to the GP k ernel.
Hence, w e w ould ha v e
( g ∗ , λ ∗ ) =argmin g ∈H k ,λ  − ln L ( D | g , λ ) − ln p ( λ ) + 1
2 k g k 2
H k  ,
where k g k 2
H k is the RKHS norm for the k ernel k . This p enalt y term can b e understo o d
as a prop er generalisation of a Gaussian log–prior densit y to function space. W e will not
giv e a formal definition here but w ork on a more heuristic lev el in the follo wing. Rather
than attempting a direct optimisation, w e will use an EM algorithm instead, applying the
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
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Donner and Opper
v ariable augmen tation with the P oisson pro cess and P´ oly a–Gamma v ariables in tro duced in
the previous sections. In this case, the lik eliho o d part of the resulting ’ Q –function’
Q (( g , λ ) | ( g , λ ) old ) .
= E P ( ω N , Π ˆ
X | ( g ,λ ) old )  ln L ( D , ω N , Π ˆ
X | g , λ )  + ln p ( λ ) − 1
2 k g k 2
H k , (30)
that needs to b e maximised in the M–step b ecomes (as in the v ariational approac h b efore)
the lik eliho o d of a Gaussian mo del in the GP function g . Hence, w e can argue that the
function g whic h maximises Q is equal to the p osterior me an of the resulting Gaussian
mo del and can b e computed without discussing the explicit form of the RKHS norm.
The conditional probabilit y measure P ( ω N , Π ˆ
X | ( g , λ ) old ) is easily obtained similar to
the optimal measure Q 1 b y not a v eraging ov er g and λ . This gives us straigh tforw ardly the
densit y
p ( ω N , Π ˆ
X | ( g , λ ) old ) = p ( ω N | ( g , λ ) old ) p (Π ˆ
X | ( g , λ ) old ) .
The first factor is
p ( ω N | ( g , λ ) old ) = p ( ω N | ( g , λ ) old )    
dP ω N
d ω N    
=
N
Y
n =1
p PG ( ω n | 1 , ˜ c n ) ,
with ˜ c n = | g old
n | . The laten t p oin t pro cess Π ˆ
X is again a P oisson pro cess densit y
p (Π ˆ
X | ( g , λ ) old ) = dP ˜
Λ
dP Λ
(Π ˆ
X | ( g , λ ) old ) ,
where the in tensit y is
˜
Λ( x , ω ) = λ old σ ( − g old ( x )) p PG ( ω | 1 , ˜ c ( x )) ,
with ˜ c ( x ) = | g old ( x ) | . The first term in the Q –function is
U ( g , λ ) .
= E P ( ω N , Π ˆ
X | ( g ,λ ) old )  ln L ( D , ω N , Π ˆ
X | g , λ ) 
= − 1
2 Z X
˜
A ( x ) g ( x ) 2 d x + Z X
˜
B ( x ) g ( x ) d x ,
with
˜
A ( x ) =
N
X
n =1
E P ( ω n | ( g ,λ ) old ) [ ω n ] δ ( x − x n ) + Z ∞
0
E P ( ω | ( g ,λ ) old ) [ ω ] ˜
Λ( x , ω ) dω ,
˜
B ( x ) = 1
2
N
X
n =1
δ ( x − x n ) − 1
2 Z ∞
0
˜
Λ( x , ω ) dω .
W e ha v e already tackled almost iden tical log–lik eliho o d expressions in Section 3.1 (see Equa-
tion (20)). While for sp ecific priors (with precision k ernels giv en b y differential operators)
an exact treatmen t in terms of solutions of ODEs or PDEs is p ossible, w e will again resort to
the sparse GP appro ximation instead. The sparse v ersion U s ( g s , λ ) is obtained b y replacing
g ( x ) → E P ( g | g s ) [ g ( x )] in U ( g , λ ). F rom this w e obtain the sparse Q –function as
Q s (( g s , λ ) | ( g s , λ ) old ) .
= U s ( g s , λ ) + ln p ( λ ) − 1
2 g >
s K − 1
s g s . (31)
12

23

Sigmoid al Ga ussian Cox Pr ocess Inference
The function v alues g s and the maximal in tensit y λ that maximise Equation (31) can b e
found analytically b y solving
∂ Q s
∂ g s
= 0 and ∂ Q s
∂ λ = 0 .
The final MAP estimate is obtained after con v ergence of the EM algorithm and the desired
sparse MAP solution for g ( x ) is giv en b y (see Equation (27))
g M AP ( x ) = k s ( x ) > K − 1
s g s
As for the v ariational sc heme, integrals o v er the space X are approximated b y Mon te–
Carlo in tegration. An alternativ e deriv ation of the sparse MAP solution can b e based on
restricting the minimisation of (30) to functions whic h are linear com binations of k ernels
cen tred at the inducing p oin ts and using the definition of the RKHS norm (see (Rasm ussen
and Williams, 2006, c hap. 6)).
Sparse Laplace p osterior T o complete the computation of the Laplace appro ximation,
w e need to ev aluate the quadratic fluctuations around the MAP solution. W e will also do
this with the previously obtained sparse appro ximation. The idea is that from the con v erged
MAP solution, w e define a sparse lik eliho o d of the P oisson mo del via the replacemen t
L s ( g s , λ ) .
= L ( D | E P ( g | g s ) [ g ] , λ )
F or this sparse lik eliho o d it is easy to compute the Laplace p osterior using second deriv a-
tiv es. Here, the change of v ariables ρ = ln λ will b e made to ensure that λ > 0. This
results in an effectiv e log–normal densit y o v er the maximal intensit y rate λ . While w e do
not address h yp erparameter selection for the Laplace p osterior in this w ork, a straigh tfor-
w ard approac h, as suggested b y Flaxman et al. (2017), could b e to use cross v alidation
to optimise the k ernel parameters while finding the MAP estimate or to use the Laplace
appro ximation to appro ximate the evidence. As in the v ariational case the inducing p oin t
lo cations { x l } L
l =1 will b e on a regular grid o v er space X .
Note that for the Laplace appro ximation, the augmen tation scheme is only used to com-
pute the MAP estimate in an efficien t w a y . There are no further mean–field appro ximations
in v olv ed. This also implies, that dep endencies b et w een g s and λ are retained.
3.3 Predictiv e densit y
Both v ariational and Laplace appro ximation yield a p osterior distribution q o v er g s and λ .
The GP appro ximation at an y giv en p oin ts in X is giv en b y
q ( g ( x )) = Z Z p ( g ( x ) | g s ) q ( g s , λ ) d g s dλ,
whic h for b oth metho ds results in a normal densit y . T o find the p osterior mean of the
in tensit y function at a p oin t x ∈ X one needs to compute
E Q [Λ( x )] = E Q  λ Z ∞
−∞
σ ( g ( x ))  .
13

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
24

Donner and Opper
F or v ariational and Laplace p osterior the exp ectation o ver λ can b e computed analytically ,
lea ving the exp ectation o ver g ( x ), whic h is computed numerically via quadrature methods.
T o ev aluate the p erformance of inference results w e are in terested in computing the likeli-
ho o d on test data D test , generated from the ground truth. W e will consider tw o metho ds:
Sampling GPs g from the p osterior w e calculate the (log) mean of the test lik eliho o d
` ( D test ) = ln E P [ L ( D test | Λ) |D ] ≈ ln E Q [ L ( D test | Λ)]
= ln E Q " exp  − Z X
λσ ( g ( x )) d x  Y
x n ∈D test
λσ ( g ( x n )) # (32)
where the in tegral in the exp onen t is approximated b y Mon te–Carlo in tegration. The ex-
p ectation is appro ximated b y av eraging o v er 2 × 10 3 samples from the inferred p osterior Q
of λ and g at the observ ations of D test and the integration points.
Instead of sampling one can also obtain an analytic appro ximation for the log test lik eliho o d
in Equation (32) b y a second order T a ylor expansion around the mean of the obtained
p osterior. Applying this idea to the v ariational mean field p osterior w e get
` ( D test ) ≈ ln L ( D test | Λ Q ) + 1
2 E Q h ( g s − µ s
2 ) > H g s   Λ Q ( g s − µ s
2 ) i
+ 1
2 H λ | Λ Q V ar Q ( λ ) ,
(33)
where Λ Q ( x ) = E Q [ λ ] σ ( E Q [ g ( x )]) and H g s   Λ Q , H λ | Λ Q are the second order deriv ativ e of
the lik eliho o d in Equation (1) with resp ect to g s and λ at Λ Q . While an appro ximation only
in v olving the first term w ould neglect the uncertain ties in the p osterior (as done b y John
and Hensman (2018)), the second and third term tak e these into accoun t.
4. Results
Generating data from the mo del T o ev aluate the t wo newly dev elop ed algorithms w e
generate data according to the sigmoidal Gaussian Co x pro cess mo del
g ∼ p GP ( ·| 0 , k ) ,
D ∼ p Λ ( · ) ,
where p Λ ( · ) is the P oisson pro cess densit y o v er sets of p oin t with Λ( x ) = λσ ( g ( x )) and
p GP ( ·| 0 , k ) is a GP densit y with mean 0 and co v ariance function k . As k ernel we c ho ose a
squared exp onen tial function
k ( x , x 0 ) = θ
d
Y
i =1
exp  − ( x i − x 0
i ) 2
2 ν 2
i  ,
where the h yp erparameters are scalar θ and length scales ν = ( ν 1 , . . . , ν d ) > . Sampling of
the inhomogeneous P oisson pro cess is done via thinning (Lewis and Shedler, 1979; Adams
et al., 2009). W e assume that h yp erparameters are known for subsequen t exp erimen ts with
data sampled from the generativ e mo del.
14

25

Sigmoid al Ga ussian Cox Pr ocess Inference
Benc hmarks for sigmoidal Gaussian Co x pro cess inference W e compare the pro-
p osed algorithms to t w o alternativ e inference metho ds for the sigmoidal Gaussian Co x
pro cess mo del. As an exact inference metho d we use the sampling approac h of Adams
et al. (2009) 4 . In terms of sp eed, a comp etitor is a different v ariational approac h giv en b y
Hensman et al. (2015) who prop osed to discretise space X in several regular bins with size
∆. Then the lik eliho o d in Equation (1) is appro ximated b y
L ( D | λσ ( g ( x ))) ≈ Y
i
p p o ( n i | λσ ( g ( x i ))∆) ,
where p p o is the P oisson distribution conditioned on the mean parameter, x i is the cen tre
of bin i , and n i the num b er of observ ations within this bin. Using a (sparse) Gaussian
v ariational appro ximation the corresp onding Kullbac k–Leibler div ergence is minimised b y
gradien t ascen t to find the optimal p osterior o v er the GP g and a p oin t estimate for λ . This
metho d w as originally prop osed for the log Co x-pro cess (Λ( x ) = e g ( x ) ), but with the elegan t
GPflo w pac k age (Matthews et al., 2017) implemen tation of the scaled sigmoid link function
is straigh tforw ard. It should b e noted, that this metho d requires n umerical integration o v er
the sigmoid link function to ev aluate the v ariational lo wer bound at every spatial bin and
ev ery gradien t step, since it do es not mak e use of our augmentation sc heme (see Section 5
for discussion, ho w the prop osed augmen tation can b e used for this mo del). W e refer to this
inference algorithm as ‘v ariational Gauss’. T o ha v e fair comparison b etw een the differen t
metho ds, the inducing p oin ts for all algorithms (except for the sampler) are equal and the
n um b er of bins used to discretise the domain X for the v ariational Gauss algorithm is set
equal to the n um b er of in tegration p oin ts used for the MC integration in the v ariational
mean field and the Laplace metho d.
Exp erimen ts on data from generativ e mo del As an illustrativ e example w e sample a
one dimensional P oisson pro cess with the generativ e mo del and p erform inference with the
sampler (2 × 10 3 samples after 10 3 burn-in iterations), the mean field algorithm, the Laplace
appro ximation and the v ariational Gauss. In Figure 1 (a) – (d) the different posterior mean
in tensit y functions with their standard deviations are sho wn. F or (b) – (d) 50 regularly
spaced inducing p oin ts are used. F or (b) – (c) 2 × 10 3 random in tegration p oin ts are dra wn
uniformly o v er the space X , while for (d) X is discretised in to the same num b er of bins. All
algorithms reco v er the true in tensit y well. The mean field and the Laplace algorithm sho w
smaller p osterior v ariance compared to the sampler. The fastest inference result is obtained
b y the Laplace algorithm in 0 . 02 s, follo w ed b y the mean field (0 . 09), v ariational Gauss
(80) and the sampler (1 . 8 × 10 3 ). The fast con v ergence of the Laplace and the v ariational
mean field algorithm is illustrated in Figure 1 (e) , where ob jectiv e functions of our t w o
algorithms (min us the maxim um they con v erged to) is shown as a function of run time.
Both algorithms reac h a plateau in only a few ( ∼ 6) iterations. T o compare p erformance in
terms of log exp ected test lik eliho o d ` test (test sets D test sampled from the ground truth),
w e a v eraged results o ver ten independent datasets. The p osterior of the sampler yields the
highest v alue with 875 . 5, while v ariational ( ` test = 686 . 2, appro ximation b y Equation (33)
yields 686 . 5), v ariational Gauss (686 . 7) and Laplace (686 . 1) yield all similar results (see also
Figure 4 (a) ). The p osterior densit y of the maximal in tensit y λ is sho wn in Figure 1 (f ) .
4. T o increase efficiency , the GP v alues g are sampled by elliptical slice sampling (Murra y et al., 2010).
15

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
26

Donner and Opper

100
200
300
(
x
)
Sampler Mean field
0 1
x
100
200
300
(
x
)
Laplace
0 1
x
Var. Gauss 0.0 0.1
Time [s]
20
10
0
Obj. func. max
Mean field
Laplace
200 400
0
1
2
-Density [×10 2 ]
Mean field
Laplace
Sampler
Var. Gauss
True
(a) (b)
(c) (d)
(e)
(f)

Figure 1: Inference on 1D dataset. (a) – (d) Inference result for sampler, m ean field
algorithm, Laplace appro ximation, and v ariational Gauss. Solid coloured lines
denote the mean in tensit y function, shaded areas mean ± standard deviation,
and dashed blac k li nes the true rate functions. V ertical bars are observ ation s D .
(e) Con v ergence of mean field and EM algorithm. Ob jectiv e functions (Lo w er
b ound for mean–field an d log lik eliho o d for EM algorithm, shifted suc h t hat
con v ergence is at 0) as function of run time (tr iangle marks one fin is h e d iteration
of the resp ectiv e algorithm). (f ) Inferred p osterior densities o v er the maximal
in tensit y λ . V ariational Gauss pro vides only a p oin t es ti m ate. Blac k v ertical bar
denotes the true λ .
16

27

Sigmoid al Ga ussian Cox Pr ocess Inference

Ground T ruth Sampler Mean field Laplace Var . Gauss
(a ) (b) (c) (d) (e)

Figure 2: Inference on 2D dataset. (a) Ground truth in tensit y function Λ( x ) with ob-
serv ed dataset D (red dots). (b) – (e) Mean p osterior in tensit y of the sampler, mean
field algorithm, Laplace, and v ariational Gauss are sho wn. 100 inducing p oin ts
on a regular grid (sho wn as coloured p oin ts) and 2500 in tegration p oin ts/bins are
used.
In Figure 2 w e sho w inference results for a t w o dimensional Co x pro cess example. 10 × 10
inducing p oin ts and 2500 in tegration p oints/bins are used for mean field, Laplace and
v ariational Gauss algorithm. The p osterior mean of sampler (b) , of the mean field (c) , of
the Laplace (d) and of the v ariational Gauss algorithm (e) reco v er the true in tensit y rate
Λ( x ) (a) w ell.
T o ev aluate the role of the n umber of inducing p oin ts and n um b er of in tegration p oin ts
w e generate 10 test sets D test from a pro cess with the same intensit y as in Figure 2 (a) .
W e ev aluate the log exp ected lik eliho o d (Equation (32)) on these test sets and compute
the a v erage. The result is sho wn for differen t n um b ers of inducing p oin ts (Figure 3 (a)
with 2500 in tegration p oin ts) and differen t n umbers of integration points (Figure 3 (b) with
10 × 10 inducing p oin ts). T o accoun t for randomness of in tegration p oin ts the fitting is
rep eated fiv e times and the shaded area is b et ween the minim um and maxim um obtained
b y these fits. F or all approximate algorithms the log predictiv e test lik eliho o d saturates
already for few inducing p oin ts ( ≈ 49 (7 × 7)) of the sparse GP . How ev er, as exp ected,
the inference appro ximations are sligh tly inferior to the sampler. The log exp ected test
lik eliho o d is hardly affected b y the num b er of in tegration p oin ts as sho wn in Figure 3 (b) .
Also the appro ximated test lik eliho o d for the mean field algorithm in Equation (33) yields
go o d estimates of the sampled v alue (dashed line in (a) and (b) ). In terms of run time
(Figure 4 (c) – (d) ) the mean field algorithm and the Laplace appro ximation are sup erior
b y more than one order of magnitude to the v ariational Gauss algorithm for this particular
example. Difference increases with increasing n um b er of inducing p oin ts.
In Figure 4 the four algorithms are compared on fiv e differen t datasets sampled from the
generativ e mo del. As w e observed for the previous examples the three differen t appro ximat-
ing algorithms yield qualitativ ely similar p erformance in terms of log test lik eliho o d ` test ,
but the sampler is sup erior. Again the appro ximated test lik eliho o d in Equation (33) (blue
star) pro vides go o d estimate of the sampled v alue. In addition w e pro vide the appro ximated
ro ot mean squared error (RMSE, ev aluated on a fine grid and normalised b y maximal in-
tensit y λ ) b et w een inferred mean and ground truth. In terms of run time the mean field
and Laplace algorithm are b y at least on order of magnitude faster than the v ariational
17

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
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Donner and Opper

145
110
115
120
test [×10 1 ]
145
110
115
120
9 25 49 81 121 169 225
Num. inducing points
10 1
10 1
10 3
Runtime [s]
900 2500 4900 8100 10000
Num. integration points
10 1
10 1
10 3
(a) (b)
(c) (d)

Figure 3: Ev aluation of inference. (a) The log exp ected predictiv e lik eliho o d a v eraged
o v er ten test sets as a function of the n um b er of inducing p oin ts. Num b er of
in tegration p oin ts/bins is fixed to 2500. Results for sampler in (red), mean fi e l d
(blue), Laplace (orange), and v ariational Gauss (purple) algorithm. S olid line
denotes mean o v er fi v e fits (same data), and shaded area denotes min. and
max. result. Dashed blue line sho ws the appro ximated log exp ec ted predictiv e
lik eliho o d for the mean field algorithm. (b) Same as (a), but as function of
n um b er of in tegration p oin ts. Num b er of inducing p oin ts is fixed to 10 × 10.
Belo w: Run time of the differen t algorithms as fu nction of n um b er of inducing
p oin ts (c) and n um b er of in tegration p oin ts (d) . Data are th e same as in Figure 2.
18

29

Sigmoid al Ga ussian Cox Pr ocess Inference

5.0
7.5
10.0
ℓ
test [× 10 −2 ]
d=1

N=166
0
5
RMSE
λ
[ 10 2 ]
S MF L VG
10 − 2
10 0
10 2
10 4
Runtime [
s
]
7.5
10.0
d=1
N=189
0
10
S MF L VG
10 − 2
10 0
10 2
10 4
10.0
12.5
15.0
d=2
N=243
0
10
S MF L VG
10 − 2
10 0
10 2
10 4
15.0
17.5
20.0
d=2
N=328
0
20
S MF L VG
10 − 2
10 0
10 2
10 4
200
210
220
d=2
N=2859
0
10
MF L VG
10 − 2
10 0
10 2
10 4
(a) (b) (c) (d) (e)

Figure 4: P erformance on differen t artificial datasets. The sampler (S), the mean
field algorithm (MF), the Laplace (L), and v ariational Gauss (V G) are compared
on fiv e differen t datasets with d –dimensions and N observ ations (one column
corresp onds to one dataset). T op ro w: Log exp ected test likelihoo d of the dif-
feren t inference results. The star denotes the appro ximated test lik eliho o d of the
v ariational algorithm. Cen ter ro w: The approximated root mean squared error
(normalised b y true maximal in tensit y rate λ ). Bottom ro w: Run time in seconds.
The dataset (e) is in tractable for the sampler due to the man y observ ations. Data
in Figure 1 and 2 corresp ond to (a) and (c) .
Gauss algorithm. In general, the mean–field algorithm seems to b e sligh tly faster than the
Laplace.
General datasets and comparison to the approac h of Llo yd et al. Next, w e test our
v ariational mean field algorithm on datasets not coming from the generativ e mo del. On such
datasets w e do not kno w, whether our mo del pro vides a go o d prior. As discussed previously
an alternativ e mo del w as prop osed b y Llo yd et al. (2015) making use of the link function
Λ( x ) = g 2 ( x ). While the sigmoidal Gaussian Co x pro cess with the prop osed augmen tation
sc heme has analytic up dates for the v ariational p osterior, in case of the squared Gaussian
Co x pro cess the lik eliho o d in tegral can b e solv ed analytically and do es not need to b e
sampled (if the k ernel is a squared exp onen tial and the domain is rectangular). Both
algorithms rely on the sparse GP appro ximation. T o compare the t w o metho ds empirically
first w e consider one dimensional data generated using a kno wn in tensit y function. W e
c ho ose Λ( x ) = 2 exp( − x/ 15) + exp( − ( x − 25) 2 / 100) on an in terv al [0 , 50] already prop osed
b y Adams et al. (2009). W e generate three training and test sets, where w e scale this
rate function b y factors of 1 , 10 , and 100 and fit the sigmoidal and squared Gaussian Co x
19

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
30

Donner and Opper

x
(
x
)
N
= 47
x
N
= 453
x
N
= 4652
(
x
) =
g
2 (
x
)
(
x
) = max (
g
(
x
))
(a) (b) (c)

Figure 5: 1D example. Observ ations (blac k bars) are sampled from the same function
(blac k line) sc aled b y (a) 1, (b) 10, and (c) 100. Blue and green line sho w the
mean p osterior of the sigmoidal and squared Gaussian Co x pro cess, resp ectiv ely .
Shaded area d e n ote s mean ± standard deviation.
Λ( x ) = λ max σ ( g ( x )) Λ( x ) = g 2 ( x )
N Run time [s] RMSE ` test Run time [ s ] RMSE ` test
47 0 . 27 ± 0 . 30 0 . 24 ± 0 . 02 − 43 . 43 ± 0 . 42 0 . 41 ± 0 . 05 0 . 24 − 44 . 26 ± 0 . 09
453 0 . 50 ± 0 . 04 0 . 97 ± 0 . 13 720 . 81 ± 0 . 28 0 . 23 ± 0 . 05 2 . 11 710 . 43 ± 1 . 38
4652 0 . 41 ± 0 . 01 7 . 68 ± 0 . 75 17497 . 31 ± 2 . 13 0 . 79 ± 0 . 09 8 . 16 17496 . 75 ± 1 . 65
T able 1: Benc hmarks for Figure 5 The mean and standard deviation of run time, RMSE,
and log exp ected test lik eliho o d for Figur e 5 (a) – (c) obtained fr om 5 fits. Note
that the RMSE f or Λ( x ) = g 2 ( x ) has no standard deviation, b ecause the in fere n c e
algorithm is deterministic.
pro cess with their corresp onding v ariational algorithm to eac h trainin g set 5 . The n um b e r
of inducing p oin ts is 40 in this example. F or our v ariational mean field algorithm w e
used 5000 in tegration p oin ts. The p osterior in tensit y Λ( x ) for the three datasets can b e
seen in Figure 5. The mo del with the sigmoidal link function infers smo other p osterior
functions with smaller v ariance compared to the p osterior with the squared link function.
F or datasets sho wn in Figur e 5 w e run the fits fiv e times and rep ort m ean and standard
deviation of ru n time, RMSE and log exp ected test li k eliho o d ` test in T able 1. Run times
of the t w o algorithms are comparable, where for the in termediate dataset the algorithm
with the squared lin k function is faster while for th e largest data set the one with the
sigmoidal link function con v erges fi rst. RMSE and ` test are also comparable except for the
in termediate dat as et, where the sigmoidal mo del is the sup erior one.
Next w e d e al with t w o real w orld t w o dimensional datasets for comparison. The first
one is neuronal d ata, where spiking activit y w as recorded from a mouse, that w as freely
mo ving in an arena (F or The Biology Of Memory and Sargolini, 2014; Sargolini et al.,
2006). Here w e consider as data D the p osition of the mou s e when t he recorded ce l l fired
and the observ ations are randomly assigned to either training or test set. In Figure 6 (a)
5. W e thank Chris Llo yd and T om G un ter for pro viding the co de for inferring the v ariational p osterior of
the squared Gaussian Co x pro cess.
20

31

Sigmoid al Ga ussian Cox Pr ocess Inference
the observ ations in the training set ( N = 583) are sho wn. In Figure 6 (b) and (c) the
v ariational p osterior’s mean in tensit y Λ( x ) is sho wn obtained for the sigmoidal and the
squared link function, resp ectiv ely , inferred with a regular grid of 20 × 20 inducing p oin ts.
As in Figure 5 w e see that the sigmoidal p osterior is the smo other one. The ma jor difference
b et w een the t w o algorithms (apart from the link function) is the fact that for the sigmoidal
mo del w e are required to sample an in terv al o v er the space. W e inv estigate the effect of the
n um b er of in tegration p oin ts in terms of run time 6 and log exp ected test lik eliho o d in Figure 6
(d) . First, w e observ e regardless of the n um b er of in tegration p oints that the v ariational
p osterior of the squared link function yields the sup erior exp ected test lik eliho o d. F or the
sigmoidal mo del the test lik eliho o d do es not impro v e significan tly with more integration
p oin ts. Run times of b oth algorithms are comparable, when 5000 integration poin ts are
c hosen. A sp eed up for our mean field algorithm is ac hiev ed b y first fitting the mo del with
1000 in tegration p oin ts and once con v erged, redra wing the desired n um b er of in tegration
p oin ts and rerun the algorithm (dotted line in Figure 6 (d) ). This metho d allows for a
significan t sp eed up without loss in terms of test lik eliho o d ` test . The v ariational mean-field
algorithm with the sigmoid link function is faster with up to 5000 in tegration p oin ts and
equally fast with 10000 in tegration p oin ts.
As second dataset w e consider the P orto taxi dataset (Moreira-Matias et al., 2013).
These data con tain tra jectories of taxi tra v els from the y ears 2013 / 14 in the city of P orto.
As John and Hensman (2018) w e consider the pic k-ups as observ ations of a P oisson pro cess 7 .
W e consider 20000 taxi rides randomly split in to training and test set ( N = 10017 and
N = 9983, resp ectiv ely). The training set is sho wn in Figure 6 (e) . Inducing p oin ts are
p ositioned on a regular grid of 20 × 20. The v ariational p osterior mean of the resp ectiv e
in tensit y is sho wn in Figure 6 (f ) and (g) . With as man y data p oin ts as in these data the
differences b et w een the t w o mo dels are more subtle as compared to (b) and (c) . In terms of
test lik eliho o d ` test the v ariational p osterior of the sigmoidal mo del (with ≥ 2000 in tegration
p oin ts) outp erforms the mo del with squared link function (Figure 6 (h) ). F or similar test
lik eliho o ds ` test our v ariational algorithm is ∼ 2 × faster than the v ariational p osterior with
squared link function. The results show that the c hoice of n um b er of integration points
reduces to the question of sp eed vs accuracy trade–off. As for the previous dataset, the
strategy of first fitting the p osterior with 1000 in tegration p oin ts and then with the desired
n um b er of in tegration p oin ts (dotted line) pro v es that w e can get a significan t sp eed up
without lo osing predictiv e p o w er.
5. Discussion and Outlo ok
Using a com bination of t w o kno wn v ariable augmentation methods, w e derive a conjugate
represen tation for the p osterior measure of a sigmoidal Gaussian Co x pro cess. The appro x-
imation of the augmen ted p osterior b y a simple mean field factorisation yields an efficient
v ariational algorithm. The rationale b ehind this metho d is that the v ariational up dates in
the conjugate mo del are explicit and analytical and do not require (blac k–b o x) gradien t
6. Note, that - in contrast to Figures 3 and 4 - the run time is displa yed on linear scale, meaning b oth
algorithms are of same order of magnitude.
7. As John and Hensman (2018) rep ort some regions to b e highly p eak ed w e consider only pic kups happ ening
within the co ordinates (41 . 147 , − 8 . 58) and (41 . 18 , − 8 . 65) in order to exclude those regions.
21

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
32

Donner and Opper

Neuronal Data
N = 583 (
x
) = max (
g
(
x
)) (
x
) =
g
2 (
x
)
50
52
54
test [×10 1 ]
1 2 5 10
Num. integration points [10 3 ]
25
50
75
Runtime [s]
Taxi Data
N = 10017 (
x
) = max (
g
(
x
)) (
x
) =
g
2 (
x
)
30
32
34
test [×10 3 ]
1 2 5 10
Num. integration points [×10 3 ]
200
300
Runtime [s]
(a) (b) (c) (d)
(e) (f) (g) (h)

Figure 6: Fits to real w orld datasets. (a) P osition of the m ou s e while the recorded neu-
ron spik ed. (b) P osterior mean obtained b y the v ari ational mean–field algorithm
for the sigmoidal Gaussian Co x pro cess. (c) Same as in (b) for the v ariational
appro ximation of the squared Gaussian Co x pro cess. (d) Log exp ected test–
lik eliho o d ` test and run time as function of n um b er of in tegration p oin ts for b oth
algorithms. The dotted line is obtained b y first fitting the sigmoidal mo del with
1000 in tegration p oin ts and then with the n um b er that is indicated on the x-axis.
Shaded area is me an ± standard deviation obtained in 5 rep eated fits. (e) – (h)
Same as (a)–(d ), but for a dataset, whe r e the observ ations are p ositions of taxi
pic k–ups in the cit y of P orto.
22

33

Sigmoid al Ga ussian Cox Pr ocess Inference
descen t metho ds. In fact, a comparison with a differen t v ariational algorithm for the same
mo del - not based on augmen tation, but on direct appro ximation of the p osterior with
a Gaussian - sho ws that the qualities of inference for b oth approac hes are similar, while
the mean field algorithm is at least one order of magnitude faster. W e use the same v ari-
able augmen tation metho d for computation of the MAP estimate for the (unaugmen ted)
p osterior b y a fast EM algorithm. This is finally applied to the calculation of Laplace’s
appro ximation. Both metho ds yield an explicit result for the appro ximate GP p osterior.
Since the corresp onding effectiv e lik eliho o d con tains a con tinuum of the GP laten t v ariables,
the exact computations of means and marginal v ariances w ould require the in v ersion of a
linear op erator instead of a simpler matrix in v erse. While for sp ecific priors, this problem
could b e solv ed b y PDE or ODE metho ds, w e resort to a w ell kno wn sparse GP approac h
with inducing p oin ts in this pap er. W e can apply this to arbitrary k ernels but need to
solv e spatial in tegrals o v er the domain. These can b e (at least for mo derate dimensionalit y)
w ell appro ximated b y simple Mon te Carlo in tegration. Adv an tage of this approac h is, that
one is not limited to rectangular domains. The only requiremen t is that the v olume |X | is
kno wn. An alternativ e P oisson mo del for whic h similar spatial integrals can be p erformed
analytically (Llo yd et al., 2015) within the sparse GP appro ximation (limited to squared
exp onen tial k ernels and rectangular domains) is based on a quadratic link function (Llo yd
et al., 2015; Flaxman et al., 2017; John and Hensman, 2018). W e compare our v ariational
algorithm with the v ariational algorithm of Llo yd et al. (2015) on different datasets and
observ e that b oth algorithms act on the same order of magnitude in terms of run time (with
sligh t adv antages for our v ariational mean field algorithm). As exp ected, w e sho w that
whether one or the other mo del is b etter in predictiv e p o w er is highly data dep enden t.
As an alternativ e to the Mon te Carlo in tegration in our approac h we could a void the
infinite dimensionalit y of the laten t GP from the b eginning b y w orking with a binning
sc heme for the P oisson observ ations as in Hensman et al. (2015). It would be straightforw ard
to adopt our augmen tation metho d to this case. The resulting P oisson lik eliho o ds w ould
then b e augmen ted b y pairs of P oisson and P´ oly a–Gamma v ariables (see Donner and Opp er
(2017)) for eac h bin. This approac h could b e fa v ourable when the num b er of observ ed
data p oin ts b ecomes v ery large, b ecause the discretisation metho d do es not scale with the
n um b er data p oin ts but with the resolution of discretisation. Ho w ev er, w e do exp ect, that
an y approac h based on either spatial discretisation or on the sparse, inducing p oin t metho d
w ould b ecome problematic for large or high dimensional domains X . Alternativ e metho ds
based on sp ectral represen tations of k ernels (Knollm ¨ uller et al., 2017; John and Hensman,
2018) are promising for tac kling those problems.
It will b e in teresting to apply the v ariable augmen tation metho d to other Ba y esian mo d-
els with the sigmoid link function. F or example, the inheren t b oundedness of the resulting
in tensit y can b e crucial for p oin t pro cesses suc h as the nonlinear Hawkes pr o c ess (Ha wk es,
1971) whic h is widely used for mo delling sto c k mark et data (Em brech ts et al., 2011) or seis-
mic activit y (Ogata, 1998). F or other p oin t pro cess mo dels the sigmoid function app ears
naturally . W e men tion the kinetic Ising mo del, a Marko v jump pro cess (Donner and Opp er,
2017) whic h w as originally in tro duced to mo del the dynamics of classical spin systems in
ph ysics. More recently it w as used to mo del the join t activit y of neurons (Dunn et al.,
2015). Finally , a Gaussian pro cess densit y mo del in tro duced b y (Murray et al., 2009) can
b e treated b y the augmen tations dev elop ed in this w ork (Donner and Opp er, 2018).
23

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
34

Donner and Opper
Ac kno wledgmen ts
CD w as supp orted b y the Deutsche F orsc h ungsgemeinsc haft (GRK1589/2) and partially
funded b y Deutsc he F orsc h ungsgemeinschaft (DF G) through gran t CR C 1294 “Data Assimi-
lation”, Pro ject (A06) “Appro ximativ e Bay esian inference and mo del selection for sto c hastic
differen tial equations (SDEs)”.
24

35

Sigmoid al Ga ussian Cox Pr ocess Inference
App endix A. P oisson pro cesses
In this paragraph w e briefly summarise those prop erties of a P oisson pro cess, whic h are
relev an t for this w ork. F or a thorough and more complete description w e recommend the
concise b o ok b y Kingman (1993), particularly c hapter 3 and 5.
W e consider a general space Z and a coun table subset Π Z = { z ; z ∈ Z } .
Definition of a P oisson pro cess A random coun table subset Π Z ⊂ Z is a P oisson
pro cess on Z , if
i) for an y sequence of disjoint subsets {Z k ⊂ Z } K
k =1 the cardinalit y of the union
N ( Z k ) .
= | { Π Z ∩ Z k } | is indep endent of N ( Z l ) for all l 6 = k .
ii) N ( Z k ) is P oisson distributed with mean R Z k Λ( z ) d z , and mean measure Λ( z ) : X → R + .
If the mean measure is constan t (Λ( z ) = Λ) the P oisson pro cess is homo gene ous , and
inhomo gene ous otherwise.
Campb ell’s Theorem Let Π Z b e a P oisson pro cess on Z with mean measure Λ( z ).
F urthermore, w e define a function h ( z ) : Z → R and the sum
H (Π Z ) = X
z ∈ Π Z
h ( z ) .
If Λ( z ) < ∞ for z ∈ Z , then
E P Λ h e ξ H (Π Z ) i = exp  Z Z  e ξ h ( z ) − 1  Λ( z ) d z  , (36)
for an y ξ ∈ C , such that the in tegral con v erges. P Λ is the probability measure of a P oisson
pro cess with in tensit y Λ( z ). Mean and v ariance are obtained as
E P Λ [ H (Π Z )] = Z Z
h ( z )Λ( z ) d z ,
V ar P Λ [ H (Π Z )] = Z Z
[ h ( z )] 2 Λ( z ) d z .
Note, that Equation (36) defines the char acteristic functional of a P oisson pro cess.
Mark ed P oisson pro cess Let Π Z = { z n } N
n =1 a P oisson pro cess on Z with intensit y Λ( z ).
Then Π ˆ
Z = { ( z n , m n ) } N
n =1 is again a P oisson pro cess on the pro duct space ˆ
Z = Z × M ,
if m n ∼ p ( m n | z n ) is dra wn indep enden tly at each z n . The m n ∈ M are the so–called
‘marks’, and the resulting Pro cess is a marke d Poisson pr o c ess with intensit y
Λ( z , m ) = Λ( z ) p ( m | z ) .
It is straigh tforw ard to extend Campb ell’s theorem and to sho w that the c haracteristic
functional of suc h a pro cess is
E P Λ h e ξ H (Π ˆ
Z ) i = exp  Z ˆ
Z  e ξ h ( z , m ) − 1  Λ( z , m ) d m d z  , (37)
with h ( z , m ) : ˆ
Z → R and H (Π ˆ
Z ) = P ( z , m ) ∈ Π ˆ
Z h ( z , m ).
25

Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
36

Donner and Opper
App endix B. The P´ oly a-Gamma densit y
The P´ oly a-Gamma densit y (P olson et al., 2013) has the useful prop ert y , that it allows to
represen t the in v erse h yp erb olic cosine by an infinite Gaussian mixture as
cosh − b ( c/ 2) = Z ∞
0
exp  − c 2
2 ω  p PG ( ω | b, 0) dω ,
with parameter b > 0. F urthermore, one can define a tilte d P´ olya-Gamma density as
p PG ( ω | b, c ) =
exp  − c 2
2 ω 
cosh − b ( c/ 2) p PG ( ω | b, 0) .
F rom those t w o equations the momen t generating function can b e obtained from the basic
definition, b eing
Z ∞
0
e ξ ω p PG ( ω | b, c ) dω = cosh b ( c/ 2)
cosh b  q c 2 / 2 − ξ
2  ,
and differen tiating with resp ect to ξ at ξ = 0 yields the first moment
E p PG [ ω ] = b
2 c tanh ( c/ 2) .
App endix C. V ariational inference for sto c hastic pro cesses
Densities for random pro cesses A sto c hastic pro cess X with probabilit y measure
P ( X ) often has no densit y with resp ect to Leb esgue measure, since X can b e an infi-
nite dimensional ob ject suc h as a function for the case of a Gaussian pro cess. Ho w ev er, one
can define densities with resp ect to another (reference) measure R ( X ) written as
p ( X ) = dP
dR ( X ) , (38)
if R ( X ) is absolutely con tin uous with resp ect to P ( X ) (if R ( X ) = 0 then P ( X ) = 0). Using
suc h a densit y , exp ectations are
E P [ f ( X )] = Z f ( X ) dP ( X ) = Z f ( x ) p ( x ) dR ( X ) = E R [ f ( x ) p ( x )] .
The densit y in Equation (38) is kno wn as the R adon–Niko d´ ym derivative of R with resp ect
to P (Konstan top oulos et al., 2011).
P oisson pro cess densit y As sp ecific example consider the prior density of the P oisson
pro cess in Equation (9), whic h is defined with resp ect to a reference measure
p Λ (Π Z ) = dP Λ
dP Λ 0
(Π Z ) = exp  − Z Z
(Λ( z ) − Λ 0 ( z )) d z  Y
z n ∈ Π Z
Λ( z n )
Λ 0 ( z n ) ,
26

37

Sigmoid al Ga ussian Cox Pr ocess Inference
where P Λ 0 is the probabilit y measure with in tensit y Λ 0 and the exp ectation is defined as
E P Λ 
 X
z n ∈ Π Z
u ( z n ) 
 = E P Λ 0 
 p Λ (Π Z ) X
z n ∈ Π Z
u ( z n ) 
 . (39)
Calculating the exp ectation of e ξ H (Π Z ) with Equation (39) w e iden tify the c haracteristic
function of a P oisson pro cess (see Equation (37)) with in tensit y Λ( z ).
Kullbac k-Leibler div ergence Using these densities w e can express the Kullbac k-Leibler
div ergence b et w een t wo probabilit y measures.
The KL–div ergence b et w een q ( X ) and p ( X ) is defined as
D KL ( Q k P ) = E Q  log dQ
dP ( X )  = Z log q ( X )
p ( X ) dQ ( X ) ,
where
q ( X ) = dQ
dR ( X ) ,
and where R ( X ) also is absolutely con tinuous to Q ( X ). The KL–div ergence do es not dep end
on the reference measure R ( X ).
App endix D. The p osterior p oin t pro cess is a mark ed P oisson pro cess
Here w e pro v e that the optimal v ariational p osterior p oin t pro cess in Equation (18) again
is a P oisson pro cess using Campb ell’s theorem. As p osterior proce ss in Equation (18) one
gets
q (Π Z ) = dQ
dP λ
(Π Z ) = Q z m ∈ Π Z e f ( z m )
E P λ h Q z m ∈ Π Z e f ( z m ) i = Q z m ∈ Π Z e f ( z m )
exp  R Z ( e f ( z ) − 1) λ ( z ) d z  ,
where Π Z is some random set of p oin ts on space Z and P λ is a random Poisson measure with
in tensit y λ ( z ). T o pro of, that the resulting p oint process density q (Π Z ) is again a P oisson
pro cess w e calculate the c haracteristic functional for some arbitrary function h : Z → R
E Q 
 Y
z m ∈ Π Z
e h ( z m ) 
 =
E P λ h Q z m ∈ Π Z e h ( z m )+ f ( z m ) i
exp  R Z ( e f ( z ) − 1) λ ( z ) d z 
= exp  R Z ( e h ( z )+ f ( z ) − 1) λ ( z ) d z 
exp  R Z ( e f ( z ) − 1) λ ( z ) d z 
= exp  Z Z
( e h ( z ) − 1) e f ( z ) λ ( z ) d z 
= exp  Z Z
( e h ( z ) − 1)Λ Q ( z ) d z  .
W e iden tify the last ro w as the generating functional of a P oisson pro cess (37) with ξ = 1.
The in tensit y of the pro cess is Λ Q ( z ) = e f ( z ) λ ( z ). With the fact that a P oisson pro cess
is uniquely c haracterised b y its generating function (Kingman, 1993, c hap. 3), the pro of is
complete.
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
38

Donner and Opper
App endix E. Sparse Gaussian pro cess appro ximation
T o solv e the inference problem for the function g , w e define a sparse GP , using the same
prior P , but by an effectiv e lik eliho o d whic h dep ends on a finite set of function v alues
g s = ( g 1 , . . . , g L ) > only . Hence, w e get
dQ s
2
dP ( g ) = q s
2 ( g s ) (40)
and the sparse p osterior measure is
dQ s
2 ( g ) = q s
2 ( g s ) dP ( g ) = dP ( g | g s ) × q s
2 ( g s ) dP ( g s ) ,
where the last equalit y holds true, since Equation (40) only dep ends on g s . The KL–
div ergence b et w een the full p osterior densit y
q 2 ( g ) = dQ 2
dP ( g ) = e U ( g )
E P  e U ( g ) 
and the sparse one q s
2 ( g s ) is giv en b y
D KL ( Q s
2 k Q 2 ) = E Q s
2  log q s
2 ( g s )
q 2 ( g )  = E P ( g s )  q s
2 ( g s ) E P ( g | g s )  log q s
2 ( g s )
e U ( g )  + const .
= E P ( g s )  q s
2 ( g s ) log q s
2 ( g s )
e E P ( g | g s ) [ U ( g )]  + const .
F rom this w e deriv e directly the p osterior densit y for the sparse GP
q s
2 ( g ) ∝ e U s ( g s ) ,
with the sparse log–lik eliho o d
U s ( g s ) = E P ( g | g s ) [ U ( g )] = Z U ( g ) dP ( g | g s ) .
28

39

Sigmoid al Ga ussian Cox Pr ocess Inference
App endix F. Lo w er b ound & h yp erparameter optimization
The lo w er b ound in Equation (12) is giv en b y
L ( q ) = E Q  log L ( D , ω N , Π ˆ
X | g , λ )
q 1 ( ω N ) q 1 (Π ˆ
X ) q s
2 ( g ) q 2 ( λ ) 
= Z ˆ
X
( E Q [ f ( ω , − g ( x ))] − E Q [log Λ 1 ] + E Q [log λ ] + 1) Λ 1 ( x , ω ) d x dω
− Z ˆ
X
Λ 1 ( x , ω ) d x dω
+
N
X
n =1 


E Q [ f ( ω n , g n )] + E Q [log λ ] − cosh c ( n )
1
2 ! +  c ( n )
1  2
2 E Q [ ω n ] 


− 1
2 tr ace ( K − 1
s (Σ s
2 + µ s
2 ( µ s
2 ) > )) − 1
2 log det(2 π K s ) + 1
2 log det(2 π e Σ s
2 )
+ α 0 log β 0 − log(Γ( α 0 )) + ( α 0 − 1) E Q [log λ ] − β 0 E Q [ λ ]
+ α 2 − log β 2 + log Γ( α 2 ) + (1 − α 2 ) ψ ( α 2 ) .
T o optimise the co v ariance k ernel parameters θ we differen tiate the lo w er b ound with re-
sp ect to these parameters and p erform then gradien t ascen t. The gradien t for one sp ecific
parameter θ is giv en b y
∂ L ( q )
∂ θ = Z ˆ
X
∂ E Q [ f ( ω , − g ( x ))]
∂ θ Λ 1 ( x , ω ) d x dω +
N
X
n =1
∂ E Q [ f ( ω n , g ( x n ))]
∂ θ
− 1
2
tr ace ( K − 1
s (Σ s
2 + µ s
2 ( µ s
2 ) > ))
∂ θ − 1
2
∂ log det(2 π K s )
∂ θ
= Z ˆ
X
∂ E Q [ f ( ω , − g ( x ))]
∂ θ Λ 1 ( x , ω ) d x dω +
N
X
n =1
∂ E Q [ f ( ω n , g ( x n ))]
∂ θ
+ 1
2 tr ace  K − 1
s
∂ K s
∂ θ K − 1
s (Σ s
2 + µ s
2 ( µ s
2 ) > ) 
− 1
2 tr ace  K − 1
s
∂ K s
∂ θ  .
The deriv ativ es of function E Q [ f ( ω , g ( x ))] are
∂ E Q [ f ( ω , g ( x ))]
∂ θ = 1
2 ∂ E Q [ g ( x )]
∂ θ − ∂ E Q  g ( x ) 2 
∂ θ E Q [ ω ] ! ,
with
∂ E Q [ g ( x )]
∂ θ = ∂ κ ( x )
∂ θ µ s
2 ,
∂ E Q  g ( x ) 2 
∂ θ = ∂ ˜
k ( x , x )
∂ θ + ∂ κ ( x )
∂ θ
>  Σ s
2 + µ s
2 ( µ s
2 ) >  κ ( x ) + κ ( x ) >  Σ s
2 + µ s
2 ( µ s
2 ) >  ∂ κ ( x )
∂ θ ,
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Chapter 3. Efficient Bayesian Infer enc e of Sigmoidal Gaussian Cox Pr o c esses
40

Donner and Opper
where κ ( x ) = k s ( x ) > K − 1
s and ˜
k ( x , x ) = k ( x , x ) − k s ( x ) K − 1
s k s ( x ) > . The remaining t w o
terms are:
∂ ˜
k ( x , x )
∂ θ = ∂ k ( x , x )
∂ θ − ∂ κ ( x )
∂ θ k s ( x ) − κ ( x ) ∂ k s ( x )
∂ θ ,
∂ κ ( x )
∂ θ = ∂ k s ( x ) >
∂ θ K − 1
s − k s ( x ) K − 1
s
∂ K s
∂ θ K − 1
s .
After eac h v ariational step the hyperparameters are up dated by
θ new = θ old + ε ∂ L ( q )
∂ θ ,
where ε is the step size.
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45

Chapter 4
Conference article: Efficient
Bayesian Infer enc e for a Gaussian
Pr o c ess Density Mo del
A ccepted for publication in the conference pro ceedings of 34
th
Confer enc e on Unc ertainty in
A rtificial Intel ligenc e (UAI) (Mon terey , United States).
Authors:
Christian Donner 1 , 2 , Manfred Opp er 1 , 2
1 T echnisc he Univ ersität Berlin. 2 Bernstein Cen ter for Computational Neuroscience Berlin.
Details:
Submitted: M arc h 2018
A ccepted: Ma y 2018
URL: h ttp://auai.org/uai2018/pro ceedings/pap ers/34.p df
arXiv URL: h ttps://arxiv.org/abs/1805.11494
License: Creativ e Commons A ttribution (CC BY 4.0)
Chapter 4
This c hapter comprises the publication (Donner and Opp er 2018a) , whic h is authored
b y myself (CD), and Prof. Manfred Opp er (MO).
Con tributions :
CD and MO conceiv ed and designed the work. CD deriv ed the inference algorithms and dev elop ed
the Python co de. CD p erformed the n umerical exp erimen ts. CD wrote the man uscript with
substan tial contribution of MO.
Python co de on GitHub: https://gith ub.com/c hristiando/SGPD_Inference.git
47

Efficient Bayesian Infer ence f or a Gaussian Pr ocess Density Model
Christian Donner ∗
Artificial Intelligence Group
T echnische Uni versität Berlin
[email protected]
Manfr ed Opper
Artificial Intelligence Group
T echnische Uni versität Berlin
[email protected]
Abstract
W e reconsider a nonparametric density model
based on Gaussian processes. By augmenting
the model with latent Pólya–Gamma random
v ariables and a latent marked Poisson process
we obtain a ne w likelihood which is conjugate
to the model’ s Gaussian process prior . The
augmented posterior allo ws for efficient infer -
ence by Gibbs sampling and an approximate
v ariational mean field approach. For the latter
we utilise sparse GP approximations to tackle
the infinite dimensionality of the problem. The
performance of both algorithms and compar -
isons with other density estimators are demon-
strated on artificial and real datasets with up to
se veral thousand data points.
1 INTR ODUCTION
Gaussian processes (GP) provide highly fle xible non-
parametric prior distrib utions ov er functions [1]. They
ha ve been successfully applied to v arious statistical prob-
lems such as e.g. regression [2], classification [3],
point processes [4] or the modelling of dynamical sys-
tems [5, 6]. Hence, it would seem natural to apply Gaus-
sian processes also to density estimation which is one of
the most basic statistical problems. GP density estima-
tion, howe ver , is a nontri vial task: T ypical realisations
of a GP do not respect non–negati vity and normalisa-
tion of a probability density . Hence, functions drawn
from a GP prior ha ve to be passed through a nonlinear
squashing function and the results ha ve to be normalised
subsequently to model a density . These operations make
the corresponding posterior distrib utions non–Gaussian.
Moreov er , likelihoods depend on all the infinitely many
∗
Also af filiated with Bernstein Center for Computational
Neuroscience.
GP function v alues in the domain rather than on the finite
number of function v alues at observed data points. Since
analytical inference is impossible, [7] introduced an in-
teresting Marko v chain Monte–Carlo sampler which al-
lo ws for (asymptotically) exact inference for a Gaussian
process density model, where the GP is passed through
a sigmoid link function. 1 The approach is able to deal
with the infinite dimensionality of the model, because
the sampling of the GP v ariables is reduced to a finite
dimensional problem by a point process representation.
Ho wev er , since the likelihood of the GP v ariables is not
conjugate to the prior , the method has to resort to a time–
consuming Metropolis–Hastings approach. In this paper
we will use recent results on representing the sigmoidal
squashing function as an infinite mixture of Gaussians in-
v olving Pólya–Gamma random v ariables [9] to augment
the model in such a way that the model becomes tractable
by a simpler Gibbs sampler . The ne w model structure al-
lo ws also for a much faster v ariational Bayesian approx-
imation.
The paper is or ganised as follo ws: Sec. 2 introduces
the GP density model, followed by an augmentation
scheme that makes its lik elihood conjugate to the GP
prior . W ith this model representation we deriv e two ef fi-
cient Bayesian inference algorithms in Sec. 3, namely an
exact Gibbs sampler and an approximate, f ast variational
Bayes algorithm. The performance of both algorithms is
demonstrated in Sec. 4 on artificial and real data. Finally ,
Sec. 5 discusses potential extensions of the model.
2 GA USSIAN PR OCESS DENSITY
MODEL
The generati ve model proposed by [7] constructs densi-
ties ov er some d -dimensional data space X to be of the
1 See [8] for an alternati ve model allo wing, howe ver , only
for approximate inference schemes.

Chapter 4. Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Density Mo del
48

form
ρ ( x | g ) = σ ( g ( x )) π ( x )
R X σ ( g ( x )) π ( x ) d x . (1)
π ( x ) defines a (bounded) base probability measure
ov er X , which is usually taken from a fix ed para-
metric family . The denominator ensures normalisation
R X ρ ( x | g ) d x = 1 . The choice of π ( x ) is important as
will be discussed Sec. 5. A prior distrib ution ov er densi-
ties is introduced by assuming a Gaussian process prior
[1] ov er the function g ( x ) : X → R . The GP is defined
by a mean function µ ( x ) (in this paper , we consider only
constant mean functions µ ( x ) = µ 0 ) and cov ariance ker -
nel k ( x , x 0 ) . Finally , σ ( z ) = 1
1+ e − z is the sigmoid func-
tion, which guarantees that the density is non–negati ve
and bounded.
In Bayesian inference, the posterior distribution of g
gi ven observed data D = { x n } N
n =1 with x ∈ X is com-
puted from the GP prior p ( g ) and the likelihood as
p ( g |D ) ∝ p ( D | g ) p ( g ) .
The likelihood is gi ven by
p ( D | g ) = Q N
n =1 σ ( g ( x n )) π ( x n )
 R X σ ( g ( x )) π ( x ) d x  N . (2)
Practical inference for this problem, ho wev er , is non-
tri vial, because (i) the posterior is non–Gaussian and (ii)
the likelihood in volv es an integral of g o ver the whole
space. Thus, in contrast to simpler problems such as GP
regression or classification, it is impossible to reduce in-
ference to finite dimensional integrals. T o circumvent the
problem that the likelihood is not conjugate to the GP
prior , [7] proposed a Metropolis-Hastings MCMC algo-
rithm for this model. W e will show in the ne xt sections
that one can augment the model with auxiliary latent ran-
dom v ariables in such a way that the resulting likelihood
is of a conjugate form allo wing for a more efficient Gibbs
sampler with explicit conditional probabilities.
2.1 LIKELIHOOD A UGMENT A TION
T o obtain a likelihood which is conjugate to the GP p ( g )
we require that it assumes a Gaussian form in g .
Repr esenting the denominator As a starting point,
we follo w [10] and use the representation
1
z N = R ∞
0 λ N − 1 e − λz dλ
Γ( N ) ,
where Γ( · ) is the gamma function. Identifying
z = R X σ ( g ( x )) π ( x ) d x in Eq. (2) we can re write the
likelihood as p ( D | g ) = R ∞
0 p ( D , λ | g ) dλ where
p ( D , λ | g ) ∝ exp  − Z X
λσ ( g ( x )) π ( x ) d x 
× p ( λ )
N
Y
n =1
λσ ( g ( x n )) π ( x n ) ,
(3)
with the improper prior p ( λ ) = λ − 1 ov er the auxiliary
latent v ariable λ . T o transform the likelihood further into
a form which is Gaussian in g , we utilise a representation
of the sigmoid function as a scale mixture of Gaussians.
Pólya–Gamma r epr esentation of sigmoid function
As discov ered by [9], the in verse hyperbolic cosine can
be represented as an infinite mixture of scaled Gaussians
cosh − b ( z / 2) = Z ∞
0
e − z 2
2 ω p PG ( ω | b, 0) dω ,
where p PG ( ω | b, 0) is the Pólya–Gamma density of ran-
dom v ariable ω ∈ R + . Moments of those densities can
be easily computed [9]. Later , we will also use the tilted
Pólya-Gamma densities defined as
p PG ( ω | b, c ) ∝ exp  − c 2
2 ω  p PG ( ω | b, 0) . (4)
These definitions allo ws for a Gaussian representation of
the sigmoid function as
σ ( z ) = e z / 2
2 cosh( z / 2) = Z ∞
0
e f ( ω ,z ) p PG ( ω | 1 , 0) dω (5)
with f ( ω , z ) = z
2 − z 2
2 ω − ln 2 . This result will be used
to transform the products ov er observations σ ( g ( x n )) in
the likelihood (3) into a Gaussian form.
W e will next deal with the first term in the likelihood
(3) which contains the integral o ver x . For this part of
the model we will deri ve a point process representation
which can be understood as a generalisation of the ap-
proach of [7].
Marked–P oisson repr esentation Utilising the sig-
moid property σ ( z )=1 − σ ( − z ) and the Pólya-Gamma
representation (5) the integral in the e xponent of Eq. (3)
can be written as a double integral
− Z X
λσ ( g ( x )) π ( x ) d x =
Z X
( σ ( − g ( x )) − 1) λπ ( x ) d x =
Z X Z R +  e f ( ω , − g ( x )) − 1  λπ ( x ) p PG ( ω | 1 , 0) dω d x

49

Next we will use a result for the characteristic function
of a Poisson process. Follo wing [11, chap. 3] one has
E φ " Y
z ∈ Π
h ( z ) # = exp  Z Z
( h ( z ) − 1) φ ( z ) d z  . (6)
h ( · ) is a function on a space Z and the expecta-
tion is ov er a Poisson process Π with rate func-
tion φ ( z ) . Π = { z m } M
m =1 denotes a random
set of points on the space Z . T o apply this re-
sult to our problem, we identify Z = X × R + ,
z = ( x , ω ) and φ λ ( x , ω ) = λπ ( x ) p PG ( ω | 1 , 0) and fi-
nally h ( z ) = e f ( ω , − g ( x )) to re write the exponential in
Eq. (3) as
e − R X λσ ( g ( x )) π ( x ) d x = E φ λ

 Y
( ω , x ) ∈ Π
e f ( ω , − g ( x )) 
 . (7)
By substituting Eq. (5) and (7) into Eq.(3) we obtain the
final augmented form of the likelihood of Eq. (2) which
is one of the main results of our paper .
p ( D , λ, Π , ω N | g ) ∝
N
Y
n =1
φ λ ( x n , ω n ) e f ( ω n ,g ( x n ))
× p φ λ (Π | λ ) p ( λ ) Y
( ω , x ) ∈ Π
e f ( ω , − g ( x )) ,
(8)
with p φ (Π | λ ) being the density ov er a Poisson process
Π = { ( x m , ω m ) } M
m =1 in the augmented space X × R +
with intensity φ λ ( x , ω ) . 2 This ne w process can be iden-
tified as a marked P oisson pr ocess [11, chap. 5], where
the e vents { x m } M
m =1 in the original data space X fol-
lo w a Poisson process with rate λπ ( x ) . Then, on each
e vent x m an independent mark ω m ∼ p PG ( ω m | b, 0) is
drawn at random from the Pólya–Gamma density . Fi-
nally , ω N = { ω n } N
n =1 is the set of latent Pólya–Gamma
v ariables which result from the sigmoid augmentation at
the observ ations x n .
A ugmented posterior over GP density W ith Eq. (8)
we obtain the joint posterior ov er the GP g , the rate scal-
ing λ , the marked Poisson process Π , and the Pólya–
Gamma v ariables at the observations ω N as
p ( ω N , Π , λ, g |D ) ∝ p ( D , ω N , Π , λ | g ) p ( g ) . (9)
In the follo wing, this ne w representation will be used to
deri ve two inference algorithms.
2 Densities such as p φ λ (Π | λ ) could be understood as the
Radon–Nykodym deri vati ve [12] of the corresponding proba-
bility measure with respect to some fixed dominating measure.
Ho we ver , we will not need an explicit form here.
3 INFERENCE
W e will first deriv e an efficient Gibbs sampler which
(asymptotically) solves the inference problem e xactly ,
and then a v ariational mean-field algorithm, which only
finds an approximate solution, b ut in a much faster time.
3.1 GIBBS SAMPLER
Gibbs sampling [13] generates samples from the poste-
rior by creating a Marko v chain, where at each time, a
block of v ariables is drawn from the conditional posterior
gi ven all the other v ariables. Hence, to perform Gibbs
sampling, we ha ve to deri ve these conditional distrib u-
tions for each set of v ariables from Eq. (9). Most of the
follo wing results are easily obtained by direct inspection.
The only non–tri vial case is the conditional distribution
ov er the latent point process Π .
Pólya-Gamma variables at obser vations The condi-
tional posterior ov er the set of Pólya–Gamma variables
ω N depends only on the function g at the observ ations
{ g ( x n ) } N
n =1 and turns out to be
p ( ω N | g ) =
N
Y
n =1
p PG ( ω n | 1 , g ( x n )) , (10)
where we ha v e used the definition of a tilted Pólya-
Gamma density in Eq. (4). This density can be efficiently
sampled by methods de veloped by [9] 3 .
Rate scaling The rate scaling λ has a conditional
Gamma density gi ven by
Gamma( λ | α, 1) = ( λ ) α − 1 e − λ
Γ( α ) . (11)
with α = | Π | + N = M + N . Hence, the posterior is
dependent on the number of observ ations and the number
on e vents of the marked Poisson process Π .
P osterior Gaussian pr ocess Due to the form of the
augmented likelihood the conditional posterior for the
GP g N + M at the observ ations { x n } N
n =1 and the latent
e vents { x m } M
m =1 is a multi variate Gaussian density
p ( g N + M | Π , ω N ) = N ( µ N + M , Σ N + M ) , (12)
with cov ariance matrix Σ N + M = [ D + K − 1
N + M ] − 1 .
The diagonal matrix D has its first N entries gi v en by
ω N follo wed by M entries being { ω m } M
m =1 . The mean
is µ N + M = Σ N + M h u + K − 1
N + M µ ( N + M )
0 i , where the
3 The sampler implemented by [14] is used for this work.

Chapter 4. Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Density Mo del
50

first N entries of N + M dimensional vector u are 1 / 2
and the rest are − 1 / 2 . K N + M is the prior cov ariance
kernel matrix of the GP e v aluated at the observed points
x n and the latent e vents x m , and µ ( N + M )
0 is an N + M
dimensional vector with all entries being µ 0 .
The predicti ve conditional posterior for the GP for any
set of points in X is simply gi v en via the conditional prior
p ( g | g N + M ) , which has a well known form and can be
found in [1].
Sampling the latent marked point pr ocess W e easily
find that the conditional posterior of the marked point
process is gi ven by
p (Π | g ,λ )= Q ω, x ∈ Π e f ( ω , − g ( x )) p φ λ (Π | λ )
exp ( R X × R + ( e f ( ω , − g ( x )) − 1 ) φ λ ( x ,ω ) dω d x ) , (13)
where the form of the normalising denominator is ob-
tained using Eq. (6). By computing the characteristic
function of this conditional point process (see App. A)
we can sho w that it is again a marked Poisson process
with intensity
Λ( x , ω ) = λπ ( x ) σ ( − g ( x )) p PG ( ω | 1 , g ( x )) . (14)
T o sample from this process we first draw Poisson
e vents x m in the original data space X using the rate
R R + Λ( x , ω ) dω = λπ ( x ) σ ( − g ( x )) [11, chap. 5]. Sub-
sequently for each e vent x m a mark ω m is generated
from the conditional density ω m ∼ p PG ( ω | 1 , g ( x m )) .
T o sample the ev ents { x m } M
m =1 , we use the well kno wn
approach of thinning [4]. W e note, that the rate is up-
per bounded by the base measure λπ ( x ) . Hence, we first
generate points ˜
x m from a Poisson process with inten-
sity λπ ( x ) . This is easily achie ved by noting that the
required number M max of such e vents is Poisson dis-
trib uted with mean parameter R X λπ ( x ) dx = λ . The
position of the e vents can then be obtained by sampling
{ ˜
x m } M max
m =1 independent points from the base density
˜
x m ∼ π ( x ) . These ev ents are thinned by keeping each
point ˜
x m with probability σ ( − g ( ˜
x m )) . The kept e vents
constitute the final set { x m } M
m =1 .
Sampling hyper parameters In this work we will con-
sider specific functional forms for the kernel k ( x , x 0 )
and the base measure π ( x ) which are parametrised by
hyperparameters θ k and θ π . These will be sampled by
a Metropolis-Hastings method [15]. The GP prior mean
µ 0 can be directly sampled from the conditional poste-
rior gi ven g M + N . In this work, the hyperparameters are
sampled e very v = 10 step. Different choices of v might
yield faster con vergence of the Mark ov Chain. Pseudo
code for the Gibbs sampler is provided in Alg. 1.
Algorithm 1: Gibbs sampler for GP density model.
Init: { x m } M
m =1 , g N + M , λ , and θ k , θ π , µ 0
1 f or Length of Markov c hain do
2 Sample PG variables at { x m } : ω N ∼ Eq. (10)
3 Sample latent P oisson pr ocess : Π ∼ Eq. (13)
4 Sample rate scaling : λ ∼ Eq. (11)
5 Sample GP : g N + M ∼ Eq. (12)
6 Sample hyper parameters : Every v th sample with
Metropolis–Hastings
7 end
3.2 V ARIA TION AL B A YES
While expected to be more ef ficient than a Metropolis-
Hastings sampler based on the unaugmented likeli-
hood [7], the Gibbs sampler is practically still limited.
The main computational bottleneck comes from the sam-
pling of the conditional Gaussian ov er function values of
g . The computation of the cov ariances requires the in-
version of matrices of dimensions N + M , with a com-
plexity O (( N + M ) 3 ) . Hence the algorithm does not
only become infeasible, when we ha v e many observ a-
tions, i.e when N is large, but also if the sampler re-
quires many thinned e vents, i.e. if M is lar ge. This can
happen in particular for bad choices of the base measure
π ( x ) . In the follo wing, we introduce a v ariational Bayes
algorithm [16], which solv es the inference problem ap-
proximately , b ut with a complexity which scales li nearly
in the data size and is independent of structure.
Structur ed mean–field appr oach The idea of vari-
ational inference [16] is to approximate an intractable
posterior p ( Z |D ) by a simpler distrib ution q ( Z ) from a
tractable family . q ( Z ) is optimised by minimising the
Kullback-Leibler di ver gence between q ( Z ) and p ( Z |D )
which is equi valent to maximising the so called varia-
tional lower bound (sometimes also called ELBO for e v-
idence lo wer bound) giv en by
L ( q ( Z )) = E Q  ln p ( Z , D )
q ( Z )  ≤ ln p ( D ) , (15)
where Q denotes the probability measure with density
q ( Z ) . A common approach for v ariational inference is
a structured mean–field method, where dependencies be-
tween sets of v ariables are neglected. For the problem at
hand we assume that
q ( ω N , Π , g , λ ) = q 1 ( ω N , Π) q 2 ( g , λ ) . (16)
A standard result for the v ariational mean–field approach
sho ws that the optimal independent factors, which max-

51

imise the lo wer bound in Eq. (15) are giv en by
ln q 1 ( ω N , Π) = E Q 2 [ln p ( D , ω N , Π , λ, g )] + const .,
(17)
ln q 2 ( g , λ ) = E Q 1 [ln p ( D , ω N , Π , λ, g )] + const . (18)
By inspecting Eq. (9), (17), and (18) it turns out that the
densities of all four sets of v ariables factorise as
q 1 ( ω N , Π) = q 1 ( ω N ) q 1 (Π) ,
q 2 ( g , λ ) = q 2 ( g ) q 2 ( λ ) .
W e will optimise the factors by a straightforward itera-
ti ve algorithm, where each factor is updated gi ven ex-
pectations ov er the others based on the previous step.
Hence, the lower bound in Eq. (15) is increased in each
step. Again we will see that the augmented likelihood in
Eq. (8) allo ws for analytic solutions of all required fac-
tors.
Pólya–Gamma variables at the obser vations Simi-
lar to the Gibbs sampler , the variational posterior of the
Pólya-Gamma v ariables at the observations is a product
of tilted Pólya–Gamma densities gi ven by
q 1 ( ω N ) =
N
Y
n =1
p PG ( ω n | 1 , c n ) , (19)
with c n = p E Q 2 [ g ( x n ) 2 ] . The only difference is, that
the second argument of p PG depends on the e xpectation
of the square of g ( x n ) .
P osterior marked P oisson process Similar to the cor-
responding result for the Gibbs sampler we can sho w 4
that the optimal latent point process Π is a Poisson pro-
cess with rate gi ven by
Λ 1 ( x , ω ) = λ 1 π ( x ) σ ( − c ( x )) p PG ( ω | 1 , c ( x ))
× e ( c ( x ) − g 1 ( x )) / 2 (20)
with λ 1 = e E Q 2 [ln λ ] , c ( x ) = p E Q 2 [ f ( x ) 2 ] , and
g 1 ( x ) = E Q 2 [ g ( x )] . Note also the similarity to the
Gibbs sampler in Eq. (14).
Optimal posterior f or rate scaling The posterior for
the rate scaling λ is a Gamma distrib ution gi ven by
q 2 ( λ ) = Gamma( λ | α 2 , 1) = λ α 2 − 1 e − λ
Γ( α 2 ) , (21)
where α 2 = N + E Q 1  P x 0 ∈ Π δ ( x − x 0 )  , and
E Q 1  P x 0 ∈ Π δ ( x − x 0 )  = R X R R + Λ 1 ( x , ω ) dω d x , and
δ ( · ) is the Dirac delta function. The integral is solv ed by
importance sampling as will be explained (see Eq. (25)).
4 The proof is similar to the one from App. A.
A pproximation of GP via sparse GP The optimal
v ariational form for the posterior g is a GP giv en by
q 2 ( g ) ∝ e U ( g ) p ( g ) ,
where U ( g ) = E Q 1 [ln p ( D , ω N , Π , λ | g )] results in the
Gaussian log–likelihood
U ( g ) = − 1
2 Z X
A ( x ) g ( x ) 2 d x + Z X
B ( x ) g ( x ) d x +const .
with
A ( x ) =
N
X
n =1
E Q 1 [ ω n ] δ ( x − x n ) + Z R +
ω Λ 1 ( x , ω ) dω ,
B ( x ) = 1
2
N
X
n =1
δ ( x − x n ) − 1
2 Z R +
Λ 1 ( x , ω ) dω .
For general GP priors, this free form optimum is in-
tractable by the fact that the lik elihood depends on g
at infinitely many points. Hence, we resort to an ad-
ditional approximation which makes the dimensionality
of the problem again finite. The well kno wn frame work
of sparse GPs [17, 18, 19] turns out to be useful in this
case. This has been introduced for likelihoods with lar ge,
b ut finite dimensional likelihoods [19, 20] and later gen-
eralised to infinite dimensional problems [21, 22]. The
sparse approximation assumes a v ariational posterior of
the form
q 2 ( g ) = p ( g | g s ) q 2 ( g s ) ,
where g s is the GP e valuated at a finite set of inducing
points { x l } L
l =1 and p ( g | g s ) is the conditional prior . A
v ariational optimisation yields
q 2 ( g s ) ∝ e U s ( g s ) p ( g s ) , (22)
where the first term can be seen as a ne w ‘effecti ve’ lik e-
lihood only depending on the inducing points. This new
(log) likelihood is gi ven by
U s ( g s ) = E P [ U ( g ) | g s ] =
− 1
2 Z X
A ( x ) ˜ g s ( x ) 2 d x + Z X
B ( x ) ˜ g s ( x ) d x + const .,
with ˜ g s ( x ) = µ 0 + k s ( x ) > K − 1
s ( g s − µ ( L )
0 ) , k s ( x )
being an L dimensional vector , where the l th entry is
k ( x , x l ) and K s being the prior cov ariance matrix for
all inducing points. The expectation is computed with
respect to the GP prior conditioned on the sparse GP g s .
W e identify Eq. (22) being a multiv ariate normal distri-
b ution with cov ariance matrix
Σ s
2 =  K − 1
s Z X
A ( x ) k s ( x ) > k s ( x ) d x K − 1
s + K − 1
s  − 1
,
(23)

Chapter 4. Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Density Mo del
52

Algorithm 2: V ariational Bayes algorithm for GP den-
sity model
Init: Inducing points, q 2 ( g s ) , q 2 ( λ ) , and θ k , θ π , µ 0
1 while L not con ver ged do
2 Update q 1
3 PG distrib utions at observ ations : q ∗
1 ( ω N )
with Eq. (19)
4 Rate of latent pr ocess : Λ 1 ( x , ω ) with Eq. (20)
5 Update q 2
6 Rate scaling : α 2 with Eq. (21)
7 Sparse GP : Σ s
2 , µ s
2 with Eq. (23), (24)
8 Update θ k , θ π , µ 0 with gradient update
9 end
and mean
µ s
2 = Σ s
2  K − 1
s Z X
k s ( x ) ˜
B ( x ) d x + K − 1
s µ ( L )
0  ,
(24)
with ˜
B ( x ) = B ( x ) − A ( x )( µ 0 − k s ( x ) > K − 1
s µ ( L )
0 ) .
Integrals ov er x The sparse GP approximation and the
posterior ov er λ in Eq. (21) requires the computation of
integrals of the form
I .
= Z X Z R +
y ( x , ω )Λ 1 ( x , ω ) dω d x ,
with specific functions y ( x , ω ) . For these functions, the
inner integral o ver ω can be computed analytically , but
the outer one ov er the space X has to be treated numeri-
cally . W e approximate it via importance sampling
I ≈ 1
R
R
X
r =1 Z R +
y ( x r , ω r ) Λ 1 ( x r , ω r )
π ( x r ) dω r , (25)
where e very sample point x r is independently drawn
from the base measure π ( x ) .
Updating hyper parameters Having an analytic so-
lution for e very factor of the v ariational posterior in
Eq. (16) we further require the optimisation of hyper -
parameters. θ k , θ π and µ 0 are optimised by maximis-
ing the lo wer bound in Eq. (15) (see App. B for explicit
form) with a gradient ascent algorithm ha ving an adap-
ti ve learning rate (Adam) [23]. Additional hyperparam-
eters are the locations of inducing points { x l } L
l =1 . Half
of them are drawn randomly from the initial base mea-
sure, while half of them are positioned on re gions with
a high density of observ ations found by a k–means al-
gorithm. Pseudo code for the complete v ariational algo-
rithm is provided in Alg. 2.
Python code for Alg. 1 and 2 is provided at [24].
4 RESUL TS
T o test our two inference algorithms, the Gibbs sampler
and the v ariational Bayes algorithm (VB), we will first
e valuate them on data dra wn from the generativ e model.
Then we compare both on an artificial dataset and se veral
real datasets. W e will only consider cases with X = R d .
T o ev aluate the quality of inference we consider alw ays
the logarithm of the e xpected test likelihood
` test ( ˜
D ) .
= ln 
 E 
 Y
x ∈ ˜
D
ρ ( x ) 
 
 ,
where ˜
D is test data unkno wn to the inference algorithm
and the expectation is o ver the inferred posterior mea-
sure. In practice we sample this expectation from the
inferred posterior ov er g . Since this quantity in v olves
an integral, that is again approximated by Eq. (25), we
check that the standard de viation std( I ) is less than 1%
of the v alue of the estimated value I .
Data from generati ve model. W e generate datasets
according to Eq. (1), where g is drawn from the GP prior
with µ 0 = 0 . As cov ariance kernel we assume a squared
exponential throughout this w ork
k ( x , x 0 ) = θ (0)
k
d
Y
i =1
exp − ( x i − x 0
i ) 2
2( θ ( i )
k ) 2 ! .
The base measure π ( x ) is a standard normal density . W e
use the algorithm described in [7] to generate exact sam-
ples. In this section, the hyperparameters θ k , θ π and µ 0
are fixed to the true v alues for inference. Unless stated
otherwise for the VB the number of inducing points is
fixed to 200 and the number of inte gration points for im-
portance sampling to 5 × 10 3 . For the Gibbs sampler ,
we sample a Marko v chain of 5 × 10 3 samples after a
b urn–in period of 2 × 10 3 samples.
In Fig. 1 we see a 1 dimensional example dataset, where
both inference algorithms recov er well the structure of
the underlying density . The inferred posterior means are
barely distinguishable. Ho we ver , ev aluating the inferred
densities on an unseen test set, we note that the Gibbs
sampler performs slightly better . Of course, this is e x-
pected since the sampler provides e xact inference for the
generati ve model and should (on a verage) not be outper -
formed by the approximate VB as long as the sampled
Marko v chain is long enough. In Fig. 1 (bottom left) we
see that only 13 iterations of the VB are required to meet
the con ver gence criterion. For Marko v chain samplers to
be ef ficient, correlations between samples should decay
quickly . Fig. 1 (bottom middle) shows the autocorrela-
tion of ` test , which was e v aluated at each sample of the

53

Figure 1: 1D data fr om the generati v e model. Data
consist of 100 samples from the underlying density sam-
pled from the GP density model. Upper left: T rue den-
sity (black line), data (black v ertical bars), mean po s te-
rior density inferred by Gibbs sampler (red dashed line)
and VB algorithm (blue line). Upper right: Ne g ati v e log
e xpected test lik elihood of Gibbs and VB inferred p os te-
rior . Lo wer left: V ariational lo wer bound as function of
iterations of the VB algorithm. Lo wer middle: Autocor -
relation of test lik elihood as function of Mark o v chain
samples obtained from Gibbs sampler . Lo wer right:
Runtime of the tw o algorithms (VB took 0 . 3 s ).
Dim # points Gibbs VB
` test T [s] ` test T [s]
1 50 -146.9 30.1 -149.2 1.13
2 100 -257.0 649.9 -260.2 2.03
2 200 -285.3 546.1 -289.6 1.41
6 400 -823.9 4667 -822.2 0.89
T able 1: P erf ormance of Gib bs s ampler and VB on dif-
ferent datasets sampled from generati v e model. ` test w as
e v aluated on a unkno wn test set including 50 samples. In
addition, runtime T is reported in seconds.
Mark o v chain. After about 10 samples the correlations
reach a plateau close to 0 , demonstrating e xcellent mix-
ing properties of the sampler . Comparing the run time
of both algorithms, VB ( 0 . 3 s ) outperforms the sampler
∼ 1 min by more than 2 orders of magnitude.
T o demonstrate the inference for more complicated prob-
lems, 2 dimensional data are generated with 200 samples
(Fig. 2). The posterior mean densities inferred by both
algorithms capture the structure well. As before, the log
e xpected test lik elihood is lar ger for the Gibbs s ampler
( ` test = − 296 . 2 ) compared to VB ( ` test = − 306 . 0 ).
Ho we v er , the Gibbs sampler took > 20 min while the
VB required only 1 . 8 s to obtain the result.
In T ab . 1 we sho w results for datasets with dif ferent size
and dif ferent dimensionality . The results confirm that the

Tr u e V B G i b b s

Figure 2: 2D data fr om generati v e model. Right: 200
samples from the underlying tw o dimensional density .
Middle: Posterior mean of Gibbs sampler inferred den-
sity . Right: Posterior mean of VB inferred density .

G i b b s V B K D E G M M

Figure 3: Comparison to other density estimati on
methods on artificial 2D data. T raining data consist of
100 data points uniformly distrib uted on a circle ( 1 . 5 ra-
dius) and additional Gaussian noise ( 0 . 2 std.). Fr om left
to right: The posterior mean inferred by Gibbs sampler
and VB algorithm, follo wed by density estimation using
KDE and GMM.
run time for the Gibbs sampler scales strongly with s ize
and dimensionality of a problem, while the VB algori thm
seems relati v ely unaf fected in this re g ard. Ho we v er , the
VB is in general outperformed by the sampler in terms of
e xpected test lik elihood or in the same range. Note, t hat
the runtime of the Gibbs sampler does not solely depend
on the number of observ ed data points N (compare data
set 2 and 3 in T ab . 1). As discussed earlier this can hap-
pen, when the base measure π ( x ) is v ery dif ferent from
the tar get density ρ ( x ) resulting in man y latent Poisson
e v ents (i.e. M is lar ge).
Cir cle data In the follo wing, we compare the GP den-
sity model and its tw o inference algorithms with tw o al-
ternati v e density estimation methods. These are gi v en by
a k ernel density estimator (KDE) with a Gaussian k e rnel
and a Gaussian mixture model (GMM) [25]. Th e free pa-
rameters of these models (k ernel bandwidth for KDE and
number of components for GMM) are optimised by 10 -
fold cross–v alidation. Furthermore, GMM is initia lised
10 times and the best result is reported. F or the GP
density model a Gaussian density is assumed as base
measure π ( x ) , and h yperparameters θ π , θ k , and µ 0 are
no w optimised. Similar to [7] we consider 100 samples
uniformly dra wn from a circle with additional Gaus sian

Chapter 4. Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Density Mo del
54

Gibbs VB KDE GMM
` test -220.31 -230.53 -228.43 -237.34
T able 2: Log expected test lik elihood f or cir cle data.

250 300 350
− ℓ t e s t
S k u l l s
G i b b s
V B
K D E
G M M

Figure 4: P erf ormance on ‘Egyptian Skulls’
dataset [26]. 100 training points and 4 dimensions.
Bar height sho ws a v erage ne g ati v e log test l ik elihood
obtained by fi v e random permutations of training and
test set and points mark single permutation results.
noise. The inferred densities (only the mean of the pos-
terior for Gibbs and VB) are sho wn in Fig. 3. Both GP
density methods reco v er well the structure of the data,
b ut the VB seems to o v erestimate the width of the Gaus-
sian noise compared to the Gibbs sampler . While the
KDE also reco v ers relati v ely well the data structure the
GMM f ails in this case. This is also reflected on the log
e xpected test lik elihoods (T ab . 2).
Real data sets The ‘Egyptian Skulls’ dataset [26] con-
tains 150 data points in 4 dimensions. 100 training points
are randomly selected and performance is e v aluated on
the remaining ones. Before fitting data is whitened. Base
measure and fitting procedure for all algorithms are the
same as for the circular data. Furthermore, fitting is
done for 5 random permutations of training and test set.
The results in Fig. 4 sho w that both algorithms for the
GP density model outperform the tw o other ones on this
dataset.
Often practical problems may consist of man y more data
points and dimensions. As discussed, the Gibbs s ampler
is not practical for such kind of problems, while the VB
could handle lar ger amounts of data. Unfortunately , the
sparsity assumption and the inte gration via i mportance
sampling is e xpected to become poorer with inc reasing
number of dimensions. Noting, ho we v er , that the ‘ef fec-
ti v e’ dimensionality in our model is determined by the
base measure π ( x ) , one can circumv ent this proble m by
an educated choice of π ( x ) if data D lie in a submanifold
of the high dimensional space X .
W e emplo y this strate gy by first fitting a GMM to the
problem and then utilising the fit as base measure. In
Fig. 5 we consider 3 dif ferent datasets 5 to test this pro-
5 Only real v alued dimensions are considered and for the

0 . 0 5
Th y r o i d
0 . 5
F i r e
0 . 0 0 . 3
A v e r a g e ℓ t e s t i m p r o v e m e n t t o G M M p e r t e s t p o i n t
W i n e

Figure 5: A pplication on higher dimensional data
with many data points. The impro v ement on log e x-
pected test lik elihood ` test per test point compared to
GMM, when using same as base measure π ( x ) for the
VB inference. Fr om top to bottom: ‘F orest Fire’
dataset [27, 28] ( 400 training points, 117 test points, 5
dim.), ‘Th yroid’ dataset [29] ( 3 × 10 3 , 772 , 6 ), ‘W ine’
dataset [27] ( 6 × 10 3 , 498 , 9 ). Bars mark impro v ement
on a v erage of random permutations of training and te st
set while points mark single runs.
cedure. As in Fig. 4, fitting is repeated 5 times for
random permutations if training and test set. F or the
‘Th yroid’ dataset, one of the 5 fits is e xcluded, be-
cause the im po r tance sampling yielded poor approxi ma-
tion std( I ) > I × 10 − 2 . The training sets contain 400 to
6000 data points with 5 to 9 dimensions. The results for
KDE are not reported, since it is al w ays outperformed by
the GMM. Fig. 5 demonstrates combining the GMM and
VB algorithm results in an impro v ement of the log t est
lik elihood ` test compared to using only GMM. A v erage
relati v e impro v ements of ` test are 8 . 9 % for ‘F orest Fire’,
4 . 1 % for ‘Th yroid’, and 1 . 1 % for ‘W ine’ dataset.
5 DISCUSSION
W e ha v e sho wn ho w inference for a nonparametric, GP
based, dens ity model can be made ef ficient. In the fol-
lo wing we w ould lik e to discuss v arious possible e xten-
sions b ut also limitations of our approach.
Choice of base measur e As we ha v e sho wn for ap-
plications to real data, the choice of the base m easure
is quite important, especially for the sampler and for
high dimensional problems. While man y datasets might
f a v our a normal distrib ution as base measure, probl ems
with outliers might f a v our f at tailed densities. In general,
an y density which can be e v aluated on the data space
X and which allo ws for ef ficient sampling, is a v alid
choice as base measure π ( x ) in our inference approach
for the GP density model. An y po werful density es tima-
‘forest fire’ dataset dimensions are e xcluded, where data ha v e
more than half 0 entries.

55

tor which fulfils this condition could provide a base mea-
sure which could then potentially be improv ed by the GP
model. It would e.g. be interesting to apply this idea to
neural networks [30, 31] based estimators. Other gen-
eralisations of our model could consider alternati ve data
spaces X . One might e.g. think of specific discrete and
structured sets X for which appropriate Gaussian pro-
cesses could be defined by suitable Mercer kernels.
Big data & high dimensionality Our proposed Gibbs
sampler suf fers from cubic scaling in the number of data
points and is found to be already impractical for prob-
lems with hundreds of observ ations. This could poten-
tially be tackled by using sparse (approximate) GP meth-
ods for the sampler (see [32] for a potential approach).
On the other hand, the proposed VB algorithm scales
only linearly with the training set size and can be ap-
plied to problems with se veral thousands of observ ations.
The integration of stochastic v ariational inference into
our method could potentially increase this limit [33].
Potential limitations of the GP density model are gi ven
by high dimensional problems. If approached nai vely ,
the combination of the sparse GP approximation and the
numerical integration using importance sampling is e x-
pected to yield bad approximations in such cases. 6 If the
data is concentrated on a lo w dimensional submanifold
of the high–dimensional space, one could still try to com-
bine our method with other density estimators providing
a base measure π ( x ) that is adapted to this submanifold,
to allo w for tractable GP inference.
Acknowledgements
CD was supported by the Deutsche F orschungsgemein-
schaft (GRK1589/2) and partially funded by Deutsche
Forschungsgemeinschaft (DFG) through grant CRC
1294 “Data Assimilation”, Project (A06) “ Approxima-
ti ve Bayesian inference and model selection for stochas-
tic dif ferential equations (SDEs)”.
Refer ences
[1] Carl Edward Rasmussen and Christopher KI
W illiams. Gaussian pr ocesses for mac hine learn-
ing , v olume 1. MIT press Cambridge, 2006.
[2] Christopher KI W illiams and Carl Edward Ras-
mussen. Gaussian processes for regression. In Ad-
vances in neural information pr ocessing systems ,
pages 514–520, 1996.
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57

A THE CONDITION AL POSTERIOR
POINT PR OCESS
Here we prov e that the conditional posterior point pro-
cess in Equation (13) again is a Poisson process using
Campbell’ s theorem [11, chap. 3]. For an arbitrary func-
tion h ( · , · ) we set H .
= P ( x ,ω ) ∈ Π h ( x , ω ) . W e calculate
the characteristic functional
E φ λ  e H   g , λ  =
E φ λ h Q ( ω , x ) ∈ Π e f ( ω , − g ( x ))+ h ( x ,ω )    g , λ i
exp  R X × R +  e f ( ω , − g ( x )) − 1  φ λ ( x , ω ) dω d x  =
exp n R X × R +  e f ( ω , − g ( x ))+ h ( x ,ω ) − 1  φ λ ( x , ω ) dω d x o
exp  R X × R +  e f ( ω , − g ( x )) − 1  φ λ ( x , ω ) dω d x  =
exp  Z X × R +  e h ( x ,ω ) − 1  e f ( ω , − g ) φ λ ( x , ω ) dω d x  =
exp  Z X × R +  e h ( x ,ω ) − 1  Λ( x , ω ) dω d x  ,
where the last equality follo ws from the definition of
φ λ ( x , ω ) and the tilted Polya–Gamma density . Using
the fact that a Poisson process is uniquely characterised
by its generating function this sho ws that the conditional
posterior p (Π | g , λ ) is a marked Poisson process.
B V ARIA TIONAL LO WER BOUND
The full v ariational lower bound is gi ven by
L ( q ) =
N
X
n =1
{ E Q [ln λ ] + ln π ( x n ) + E Q [ f ( ω n , g ( x n ))]
− ln cosh  c n
2  + c 2
n
2 E Q [ ω n ] 
+ Z X Z R +
{ E Q [ln λ ] + E Q [ f ( ω , − g ( x ))] − ln λ 1
− ln σ ( − c ( x )) − ln cosh  c ( x )
2  − c ( x ) 2
2 ω
− c ( x ) − g 1 ( x )
2 + 1  Λ 1 ( x , ω ) dω d x
− E Q [ λ ] + E Q  ln p ( λ )
q ( λ )  + E Q  ln p ( g s )
q ( g s )  .

Chapter 4. Efficient Bayesian Infer enc e for a Gaussian Pr o c ess Density Mo del
58

Chapter 5
Journal article: Inverse Ising
pr oblem in c ontinuous time: A
latent variable appr o ach
Published in the journal Physic al R eview E (American Ph ysical So ciet y , United States).
Authors:
Christian Donner 1 , 2 , Manfred Opp er 1 , 2
1 : T echnisc he Univ ersität Berlin. 2 : Bernstein Center for Computational Neuroscience Berlin.
Details:
Submitted: S eptem b er 2016
A ccepted: Ma y 2017
DOI: h ttps://doi.org/10.1103/PhysRevE.96.062104
Pubmed-ID: 29347355
License: Reprin ted with p ermission from [Christian Donner & Manfred Opp er, Physical Review E,
96, 062104, 2017] Cop yright (2017) b y the American Ph ysical So ciet y .
Python co de on GitHub: https://gith ub.com/c hristiando/dynamic_ising
59

PHYSICAL REVIEW E 96 , 062104 (2017)
In v erse Ising pr oblem in continuous time: A latent variable appr oach
Christian Donner * and Manfred Opper
Artificial Intelligence Gr oup, T echnisc he Universität, Mar chstr . 23, 10587 Berlin, Germany
(Recei ved 1 September 2017; published 4 December 2017; corrected 27 December 2017)
W e consider the in verse Ising problem: the inference of netw ork couplings from observed spin trajectories for
a model with continuous time Glauber dynamics. By introducing two sets of auxiliary latent random v ariables
we render the likelihood into a form which allo ws for simple iterativ e inference algorithms with analytical
updates. The v ariables are (1) Poisson v ariables to linearize an exponential term which is typical for point process
likelihoods and (2) Pólya-Gamma v ariables, which make the likelihood quadratic in the coupling parameters.
Using the augmented likelihood, we deri ve an expectation-maximization (EM) algorithm to obtain the maximum
likelihood estimate of network parameters. Using a third set of latent v ariables we extend the EM algorithm to
sparse couplings via L1 regularization. Finally , we dev elop an ef ficient approximate Bayesian inference algorithm
using a v ariational approach. W e demonstrate the performance of our algorithms on data simulated from an Ising
model. For data which are simulated from a more biologically plausible netw ork with spiking neurons, we sho w
that the Ising model captures well the lo w order statistics of the data and ho w the Ising couplings are related to
the underlying synaptic structure of the simulated network.
DOI: 10.1103/PhysRe vE.96.062104
I. INTR ODUCTION
In recent years, the in verse Ising problem, i.e., the recon-
struction of couplings and external fields of an Ising model
from samples of spin configurations, has attracted considerable
interest in the physics community [ 1 ]. This is due to the fact
that Ising models play an important role for data modeling
with applications to neural spike data [ 2 , 3 ], protein structure
determination [ 4 ], and gene expression analysis [ 5 ]. Much
ef fort has been dev oted to the dev elopment of algorithms for
the static in verse Ising problem. This is a nontri vial task, be-
cause statistically ef ficient, likelihood-based methods become
computationally infeasible by the intractability of the partition
function of the model. Hence one has to resort to either
approximate inference methods or to other statistical estima-
tors such as pseudolikelihood methods [ 6 ] or the interaction
screening algorithm [ 7 ]. The situation is some what simpler for
the dynamical in verse Ising problem, which recently attracted
attention [ 8 – 13 ]. If one assumes a Marko vian dynamics, the
exact normalization of the spin transition probabilities allo ws
for an explicit computation of the likelihood if one has a
complete set of observed data o ver time. Ne vertheless, the
model parameters enter the likelihood in a fairly comple x way ,
and the application of more adv anced statistical approaches
such as Bayesian inference again becomes a nontri vial task.
This is especially true for the continuous time kinetic Ising
model where the spins are gov erned by Glauber dynamics
[ 14 ]. W ith this dynamics the likelihood contains an e xponential
function related to the “nonflipping” times and makes analyt-
ical manipulations of the posterior distribution of parameters
intractable. Ho wev er , it is possible to compute the likelihood
gradient to find the maximum likelihood estimate (MLE) [ 15 ].
In this paper we will sho w ho w the likelihood for the
continuous time problem can be remarkably simplified by
introducing a combination of two sets of auxiliary random
* Also at Bernstein Center for Computational Neuroscience;
[email protected]
v ariables. The first set of v ariables are Poisson random v ari-
ables which “linearize” the aforementioned exponential term
that appears naturally in likelihoods of Poisson point-process
models [ 16 ]. These latent v ariables are related to previous
work, where similar v ariables ha ve been introduced for
sampling the intensity function of an inhomogeneous Poisson
process [ 17 ]. The second set of v ariables are the so-called
Pólya-Gamma v ariables, which were introduced into statistics
to enable ef ficient Bayesian inference for logistic regression
[ 18 ] and which may not be familiar in the ph ysics community .
These v ariables hav e also been used recently for Monte Carlo–
based Bayesian inference of discrete-time Marko v models
[ 19 ], model-based statistical testing of spike synchron y [ 20 ],
and an expectation-maximization (EM) scheme for logistic
regression [ 21 ].
W ith these latent v ariables the model parameters enter the
resulting joint likelihood similarly to simple Gaussian models.
W e will use this formulation to construct iterativ e algorithms
for a penalized maximum likelihood and for v ariational
Bayes estimators which ha ve simple analytically computable
updates. W e test our algorithms on artificial data. As an
illustrati ve application we use the Bayes algorithm on data
from a simulated recurrent network with conductance-based
spiking neurons and sho w how the model reproduces the
statistics of the data and ho w the obtained Ising parameters
reflect the underlying synaptic structure.
The paper is org anized as follo ws: In Sec. II the continuous
time kinetic Ising model is introduced follo wed by a deriv ation
of its likelihood in Sec. III . In Sec. IV we introduce auxiliary
latent v ariables to simplify the likelihood. In Sec. V we
de velop an EM algorithm for maximum likelihood inference
and extend it to L1-re gularized likelihood maximization and a
v ariational Bayes approximation. Finally , in Sec. VI we apply
our method to simulated data generated from an Ising network
and from a network of spiking neurons.
II. THE MODEL
Follo wing Ref. [ 15 ] in this section, we consider a system
of N Ising spins s i ( t ) ∈{ − 1 , 1 } for i = 1 ,..., N . W e denote
2470-0045/2017/96(6)/062104(9) 062104-1 ©2017 American Physical Society

Chapter 5. Inverse Ising pr oblem in c ontinuous time: A latent variable appr o ach
60

CHRISTIAN DONNER AND MANFRED OPPER PHYSICAL REVIEW E 96 , 062104 (2017)
the vector of all spins by s ( t ) = ( s 1 ( t ) ,..., s
N ( t ) )  .As p i n i
is interacting with spin j through a coupling J ij . W e are not
assuming symmetry of these couplings: in general, we hav e
J ij = J ji . W e will also allo w for self-couplings J ii . The total
field acting on spin i is gi ven by
H i ( t ) = θ i +
N

j = 1
J ij s j ( t ) , (1)
where θ i denotes the e xternal field . The Glauber dynamics of
the spins is defined by asynchronous updates [ 15 ] where in
a small time interv al t , spins i are selected independently
with probability γ t for an update; γ> 0 is the update rate.
The updated spins are flipped, i.e., s i ( t + t ) =− s i ( t ) with
probability
P flip
i ( t ) = exp [ − s i ( t ) H i ( t )]
2 cosh[ H i ( t )] . (2)
The probability that spin i is not flipped at time t in the interval
t is giv en by 1 − γ t + γ t [1 − P flip
i ( t )] . Hence, the total
probability of a (time-discretized) temporal sequence { s } 0: T of
spins during a time interv al [0 : T ]i sg i v e nb y
P ( { s } 0: T | J ) = 
( i,t ) ∈ F  γ t ex p[ − s i ( t ) H i ( t )]
2 cosh[ H i ( t )] 
× 
( i,t ) ∈ NF  1 − γ t + γ t exp [ s i ( t ) H i ( t )]
2 cosh[ H i ( t )]  .
(3)
Here F denotes the set of pairs ( i,t ) where spin i was flipped
at time t , and NF is the corresponding, complementary set
of times and spins where no flips happened. J stands for the
parameters of the model: J ≡ J ij for i,j = 1 ,...N and θ i for
i = 1 ,..., N .
III. LIKELIHOOD AND INFERENCE
Our goal is to infer the couplings and external fields from
observ ations of complete spin trajectories ov er a time interv al
[0 ,T ]. W e will consider only likelihood- based approaches
in this paper . Hence, we need to compute the probability
of spin trajectories ( 3 ) as a function of parameters, i.e., the
so-called likelihood function in continuous time. T aking the
limit t → 0i n( 3 ) and discarding pref actors which contain
t b ut are irrele vant for inference (being independent of J ),
the complete-data likelihood function [ 16 ] is found to be
L ( { s } 0: T | J ) = 
( i,t ) ∈ F
exp [ − s i ( t ) H i ( t )]
2 cosh[ H i ( t )]
×
N

i = 1
exp  γ  T
0  exp [ s i ( t ) H i ( t )]
2 cosh[ H i ( t )] − 1  dt  .
(4)
A maximum likelihood estimate of the parameters J can be
obtained by a (possibly penalised) gradient ascent approach
of this function [ 15 ]. Ho we ver , a Bayesian inference approach
does not seem to be feasible from the expression ( 4 ). F or a
Bayesian approach one would introduce a prior density p ( J )
of parameters and would infer statistical properties of J using
the posterior density gi v en by
p ( J |{ s } 0: T ) = L ( { s } 0: T | J ) p ( J )
 L ( { s } 0: T | J ) p ( J ) d J , (5)
from which posterior expectations of parameters w ould ha ve
to be calculated by high-dimensional integrals. Due to the
complex dependenc y of the likelihood on the parameters,
the application of well-kno wn techniques such as Monte
Carlo sampling, e.g., using a Gibbs sampler , or approximate
inference methods such as the v ariational approach [ 22 ]
would not be tri vial. W e will sho w in the next section that
the dependency of the lik elihood on J can be remarkably
simplified by augmenting the system by two sets of auxiliary
random v ariables.
IV . V ARIABLE A UGMENT A TION AND
TRA CT ABLE LIKELIHOOD
The two main problems that pre vent us from performing
ef ficient analytical inference using Eq. ( 4 ) come from two
sources: first, the time integral which contains the parameters J
appears in an exponential function, and, second, the parameters
also appear in the denominators in the hyperbolic cosine
function. W e will show that both problems can be solv ed by
the introduction of auxiliary v ariables. W e will start with a
simplification of the integral.
A. Poisson variables
W e note that fields H i ( t ) are piecewise constant functions
of time and do not change where no spin is flipped. Hence, the
time integral can be calculated analytically . W e will order the
constant interv als and number them by n ∈{ 0 , 1 ,..., n
max } .
W e define H n
i and s n
i as the v alues of the field and spin i
between time points t n and t n + 1 . t n denotes the time of the n th
flip time for n ∈{ 1 ,..., n
max } , while t 0 = 0 and t n max + 1 = T .
Hence, we obtain
 T
0
exp [ s i ( t ) H i ( t )]
2 cosh[ H i ( t )] dt =
n max

n = 0
exp  s n
i H n
i 
2 cosh  H n
i  ( t n + 1 − t n ) . (6)
Introducing a set of independent Poisson distributed random
v ariables ρ n
i for each i and each time slice between t n + 1 and
t n , we obtain the follo wing representation of the second part
of the likelihood:
exp  γ  T
0  exp [ s i ( t ) H i ( t )]
2 cosh[ H i ( t )] − 1  dt 
=
n max

n = 0
⎧
⎨
⎩
∞

ρ n
i = 0  exp  s n
i H n
i 
2 cosh  H n
i   ρ n
i
P Po  ρ n
i   γ ( t n + 1 − t n )  ⎫
⎬
⎭
,
(7)
where
P Po ( ρ | ζ ) = e − ζ ζ ρ
ρ ! (8)
062104-2

61

INVERSE ISING PR OBLEM IN CONTINUOUS TIME: A . . . PHYSICAL REVIEW E 96 , 062104 (2017)
denotes a Poisson distrib ution with mean parameter ζ .F o r
Eq. ( 7 ) we made use of the equality
e ζ ( x − 1) = ∞

ρ = 0
x ρ P Po ( ρ | ζ ) ,
which is the moment-generating function of the Poisson
distrib ution [ 23 ]. Similar v ariables were used in Ref. [ 17 ]t o
make Poisson-process lik elihoods tractable for Monte Carlo
sampling.
B. Pólya-Gamma variables
T o get rid of the hyperbolic terms in the denominators, we
will use a remarkable representation which was disco vered
and used in the statistics literature in recent years to simplify
Bayesian inference for logistic regression. Reference [ 18 ]
found a con venient form of writing an in verse hyperbolic
cosine as a continuous mixture of Gaussian densities as
cosh − b ( x ) =  ∞
0
dω e − 2 ωx 2 p PG ( ω | b, 0) , (9)
where p PG ( ω | b, 0) is the Pólya-Gamma density with parameter
b . Surprisingly , the exact form of this distrib ution is not of
importance for our inference algorithm, b ut only the fact
that one can deri ve its first moments straightforwardly (see
Appendix B ). Introducing Pólya-Gamma v ariables ω into the
likelihood ( 7 ) yields the representation
p ( { s } 0: T | J ) = 
ρ  L ( { s , ρ , ω } 0: T | J ) d ω , (10)
with the augmented likelihood
L ( { s , ρ , ω } 0: T | J )
= 
( i,t ) ∈ F
exp {− s i ( t ) H i ( t ) − 2[ H i ( t )] 2 ω i ( t ) } p PG ( ω i ( t ) | 1 , 0)
× 
i,n
 exp  ρ n
i  s n
i H n
i − ln(2)  − 2  H n
i  2 ω n
i 
× P Po  ρ n
i   γ ( t n + 1 − t n )  p PG  ω n
i   ρ n
i , 0   . (11)
The adv antage of the augmented likelihood ov er the original
one is the fact that the parameters appear at most quadratically
in the exponential functions [note that the fields H i ( t ) are linear
functions of the parameters]. As we will see, the computation
of maximum likelihood and related estimators as well as
Bayesian inference become considerably facilitated. W e will
postpone explicit results of Gibbs sampling algorithms to a
future publication and discuss applications of the augmented
likelihood to penalized maximum lik elihood estimation and to
a v ariational Bayes algorithm in this paper .
V . INFERENCE
A. EM algorithm
The EM algorithm [ 24 ] is a con venient w ay to maximize
the likelihood iterati vely with respect to J by using latent
v ariable representations. The algorithm cycles between an E
step and an M step and guarantees to increase the likelihood
( 4 ) in each step. At iteration m + 1, in the Es t e p one computes
the cost function Q ( J , J m ). It equals the expectation of the
logarithm of the augmented likelihood with respect to the
distrib ution of latent v ariables conditioned on the parameters
at the pre vious iteration m :
Q ( J , J m ) .
= 
ρ  d ω p ( ρ , ω |{ s } 0: T , J m )
× ln L ( { s , ρ , ω } 0: T | J ) . (12)
For t he Ms t e p we compute an update of the parameters via
J m + 1 = arg max
J Q ( J , J m ) . (13)
The conditional distribution is gi ven by
p ( { ρ , ω } 0: T |{ s } 0: T , J )
= p ( { ω } 0: T |{ s , ρ } 0: T , J ) P ( { ρ } 0: T | J , { s } 0: T ) , (14)
where
p ( { ω } 0: T |{ s , ρ } 0: T , J )
= 
( i,t ) ∈ F
p PG ( ω i ( t ) | 1 , 2 H i ( t ) ) 
n,i
p PG  ω n
i   ρ n
i , 2 H n
i  , (15)
where we defined the tilted Pólya-Gamma distribu tion as
p PG  ω n
i   b,c  = exp  − c 2
2 ω n
i  p PG  ω n
i   b, 0 
cosh − b  c
2  ,
and where
P ( ρ | J , { s } 0: T ) = 
n,i
P Po  ρ n
i    
γ ( t n + 1 − t n ) exp  s n
i H n
i 
2 cosh  H n
i   .
(16)
The first part of the conditional density is ov er factorizing
Pólya-Gamma v ariables and the second one over f actorizing
Poisson random v ariables. The necessary expectations for
the E step follo w from simple properties of Poisson random
v ariables and of Pólya-Gamma random variables deri ved in
Appendix B . This results in
 ω i ( t ) = 1
4 H i ( t ) tanh[ H i ( t )] ,
 ω n
i  =  ρ n
i 
4 H n
i
tanh  H n
i  ,
 ρ n
i  = ( t n + 1 − t n ) γ exp  s n
i H n
i 
2 cosh  H n
i  ,
(17)
where the brackets · denote expectations conditioned on J m .
Since the augmented log-likelihood is a quadratic form in the
parameters J , the maximization leads to N systems of linear
equations for the vectors J i ·
.
= ( θ i ,J i 1 ,..., J
iN )  of the form
A i J i · = b i · (18)
with
b ij =− 
t ∈ F ( i )
s i ( t ) s j ( t ) + 
n  ρ n
i  s n
i s n
j (19)
and
A ij k = 4 ⎡
⎣ 
t ∈ F ( i )  ω t
i  s k ( t ) s j ( t ) + 
n  ω n
i  s n
k s n
j ⎤
⎦ . (20)
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CHRISTIAN DONNER AND MANFRED OPPER PHYSICAL REVIEW E 96 , 062104 (2017)
Here F ( i ) is the set of all times that spin i has flipped. As
mentioned before, only the first moment of the Pólya-Gamma
density is required.
B. Sparsity via L1 regularization
Assuming a factorizing Laplace distrib ution ov er each
coupling J ij ,
p ( J ij ) = λ
2 exp ( − λ | J ij | ) ,
will enforce sparsity on the network. λ is the scale parameter
of this density . On the le v el of the MAP (maximum a poste-
riori ) Bayesian estimator this is equi v alent to L1 regularized
maximum likelihood estimation. Ho wev er , the absolute value
in the exponent of this prior w ould pre vent us from using the
pre viously described EM procedure directly and allow only for
gradient methods similar to Ref. [ 25 ]. Fortunately , this problem
can again be solv ed by the introduction of a further auxiliary
random v ariable for each single coupling parameter J ij .T h i s
follo ws from the fact that a Laplace distribution can once more
be represented as an infinite mixture of Gaussians [ 26 , 27 ],
λ
2 exp ( − λ | J | ) =  dβ  βλ 2
2 π exp  − βλ 2
2 J 2  p ( β ) , (21)
with
p ( β ) = ( β/ 2) − 2 exp [ − 1 / (2 β ) ] .
By extending the augmented lik elihood ( 11 )t o spar sity
variables { β ij } a similar EM algorithm is possible to obtain
the L1-regularized ML solution of J . The required conditional
density factorizes as
p ( { ρ , ω } 0: T , β |{ s } 0: T , J ) = p ( { ρ , ω } 0: T |{ s } 0: T , J ) p ( β | J ) ,
(22)
where p ( β | J ) = i,j p ( β ij | J ij ) and each factor is a
gener alized in verse Gaussian distrib ution
p ( β ij | J ij ) = p GIG ( β ij | a ij , 1 ,ν )
= a ν/ 2
ij
2 K ν ( √ a ij ) β ν − 1
ij exp  − a ij β ij + 1 /β ij
2  , (23)
where a ij = λ 2 J 2
ij , ν =− 1 / 2, and K ν is the modified Bessel
function of the second kind. The only change in the linear
system ( 18 ) is in the matrices A , which hav e to be replaced by
A sparse
ij k = A ij k + δ i,k λ 2  β ij  , (24)
and where  β ij = ( J 2
ij λ 2 ) − 1 / 2 (see Appendix C ).
C. Appr oximate posterior distribution via variational Bay es
For Bayesian inference we assume the pre viously discussed
Laplace prior ov er couplings J ij with scaling parameter λ and
for the external fields θ i a Gaussian prior with mean μ θ and
precision λ 2
θ . T o obtain a full posterior distribution including
the couplings J we could either sample from the posterior or
resort to a v ariational approach. The latter method is popular in
the field of machine learning [ 22 ] b ut has its roots in statistical
physics [ 28 ]. In our case we assume approximated posterior
that has the follo wing factorizing form:
p ( J , { ω , ρ } 0: T , β |{ s } 0: T ) ≈ q ( J , { ω , ρ } 0: T , β )
≡ q 1 ( J ) q 2 ( { ω , ρ } 0: T , β ) , (25)
where the two f actors q 1 and q 2 are optimized to minimize the
relati ve entropy (K ullback-Leibler) div er gence:
D ( q ; p ) = 
ρ   q ( J , { ω , ρ } 0: T , β )
× ln q ( J , { ω , ρ } 0: T , β )
p ( J , { ω , ρ } 0: T , β |{ s } 0: T ) d ω d β d J  . (26)
This is equi v alent to minimizing the variational fr ee ener gy :
F ( q ; p ) = 
ρ   q ( J , { ω , ρ } 0: T , β )
× ln q ( J , { ω , ρ } 0: T , β )
p ( { s , ω , ρ } 0: T , J , β ) d ω d β d J  . (27)
The neg ati ve free energy is actually a lower bound on the log
marginal lik elihood
− F ( q ; p ) ⩽  L ( { s } 0: T | J ) p ( J ) d J , (28)
and can be used directly for approximate model selection
[ 22 ], while in a pure maximum likelihood approach this is
not possible.
Minimizing the v ariational free energy with respect to the
factors of our f actorizing distrib ution ( 25 ), the optimal factors
turn out to be
q 
1 ( J ) ∝ exp   ln p ( J , { s , ω , ρ } 0: T , β )  q 2  ,
q 
2 ( { ω , ρ } 0: T , β ) ∝ exp   ln p ( J , { s , ω , ρ } 0: T , β )  q 1  ,
which are obtained by iterati v e updates [ 22 ]. For the posterior
at hand we find the optimal factor q 2 of the posterior
q 
2 ( ρ , ω , β ) = 
( i,t ) ∈ F
q 2 ( ω i ( t )) 
i,n
q 2  ω n
i   ρ n
i  q 2  ρ n
i  q 2 ( β )
= 
( i,t ) ∈ F
p PG ( ω i ( t ) | 1 , 2 !  [ H i ( t )] 2  )
× 
i,n
p PG  ω n
i   ρ n
i , 2 "  H n
i  2 
× P Po ⎧
⎨
⎩
ρ n
i    
γ ( t n + 1 − t n ) exp  s n
i  H n
i 
2 cosh  "  H n
i  2 
⎫
⎬
⎭
× 
( ij )
p GIG  β ij    J 2
ij  λ 2 , 1 , − 1 / 2  . (29)
From the fact that the augmented likelihood ( 11 ) and the
sparsity prior factorize in the components J i · it follows that the
optimal posterior q 
1 ( J ) does so as well. Each of those factors
is a Gaussian distrib ution with cov ariance and mean gi v en by
 i = [4 A i + ( ˜
 i ) − 1 ] − 1 , (30)
μ i =  i [ b i + ( ˜
 i ) − 1 ˜
μ i ] , (31)
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INVERSE ISING PR OBLEM IN CONTINUOUS TIME: A . . . PHYSICAL REVIEW E 96 , 062104 (2017)
− 0 . 2 0 . 00 . 2
J
− 0 . 2
0 . 0
0 . 2
J est
02468
EM Iteration
− 35
− 34
− 33
ln L [10 5 ]
0 . 30 . 50 . 70 . 91 . 11 . 3
g
10 − 4
10 − 3
MSE
10 1 10 2 10 3
T
10 − 4
10 − 3
MSE
(a) (b)
(c) (d)
FIG. 1. Inference with EM algorithm on artificial data. (a) T rue
couplings (black dots) and external fields (red triangles) vs inferred
ones. (b) The log-likelihood as function of EM iterations. The
parameters are set to N = 40, T = 10 3 ,a n d g = 0 . 3 with external
fields θ = 0 . (c) MSE between J and J est as a function of scaling
factor of the v ariance g and (d) as a function of data length T . If not
changed parameters are as in (a).
where ˜
 i is a diagonal matrix with dia g ( ˜
 − 1
i ) =
( λ 2
θ ,λ 2  β i 1  ,..., λ
2  β iN  ). The prior mean is defined as ˜
μ i =
( μ θ , 0 ,..., 0)  . Similar to the EM algorithm, we hav e a
v ariational step, where q 2 is optimized, gi ven q 1 and a second
one, optimizing q 1 gi ven q 2 . The v ariational step updating q 2
dif fers from E step in the sense that here expectations ov er
the terms with the couplings J are required and not only the
pointwise estimate (see Appendix D ). The v ariational M step
is similar to the EM algorithm, where the expectations for A
and b are computed with respect to q 2 .
The Python code of the algorithms discussed here is
publicly a v ailable [ 29 ].
VI. RESUL TS
W e test the EM algorithm on artificial data generated with
random couplings J ij from a Gaussian distrib ution with mean
0 and v ariance g 2 /N , where scaling factor g = 0 . 3. W ith
external fields θ = 0 and update rate γ = 100 data is generated
with a Gillespie algorithm [ 16 ] (see Appendix A ).
A. Maximum likelihood
In Fig. 1 the inference results for the EM algorithm are
sho wn. Figures 1(a) and 1(b) present a single fit with N = 40
spins and data length T = 10 3 . The inferred couplings J est
agree well with the true couplings J . The logarithm of the
likelihood ( 4 ) con verges well after eight EM iterations. The
mean squared error (MSE) increases with increasing scaling
02 0 4 0
λ
0
− 20
− 40
− 60
Δl n L test ( λ )
02 0 4 0
λ Ba ye s
0
5
10
Δ F ( λ Ba ye s )[10 2 ]
0 . 00 . 51 . 0
F alse P ositive Rate
0 . 0
0 . 5
1 . 0
T rue P ositive Rate
02 0 4 0
λ
0 . 80
0 . 81
0 . 82
AU C
) b ( ) a (
) d ( ) c (
FIG. 2. Inference of sparse couplings with EM and variational
Bayes. Artificial data ( T = 50 ,N = 25 ,g = 0 . 3) are generated, but
each coupling is set to 0 with probability p sparse = 1 / 2. (a) Dif ference
in likelihood (with respect to the lik elihood obtained with optimal λ  )
of couplings J est inferred by EM as a function of regularization pa-
rameter λ . Likelihood L test is computed on unseen test data ( T = 50).
The optimal parameter is λ  = 29 . 4 (red diamond). The vertical line
marks the v ariational estimation λ 
Bayes . (b) Difference in free ener gy
F (with respect to the likelihood obtained with estimate of optimal
λ 
Bayes ) of the variational Bayes algorithm. The optimal parameter is
λ 
Bayes = 34 . 5 (blue diamond). (c) R OC curves for the λ  (EM, solid
red line) and λ 
Bayes (Bayes, dashed blue line), respecti vely . (d) The
A UC for different parameters λ for the EM result (solid black line
with squares) and the v ariational Bayes algorithm (dashed gray line
with triangles). Diamonds mark the optimal λ  and the estimate λ 
Bayes .
of the coupling v ariance g [Fig. 1(c) ] and decreases linearly
on a log-log scale with increasing data length T [Fig. 1(d) ].
B. L1 regularization and variational Bay es
Regularization becomes particular important once little
data are at hand. T o test this we generate couplings as
before for a network of N = 25 spins, b ut a coupling is set
to 0 with probability of p sparse = 0 . 5. Generated data hav e
length T = 50. W e run the L1-regularized EM algorithm with
dif ferent values of λ and define the optimal λ  , whose MLE J est
maximizes the likelihood L test on unseen test data ( T = 50)
generated by the true Ising parameters J [see Fig. 2(a) ]. For
inference by the v ariational algorithm on the same training
data we estimate the optimal λ 
Bayes by taking the v alue that
minimizes the free energy ( 27 ) [Fig. 2(b) ]. Note that the
Bayesian algorithm requires no test data for this estimate.
Next we try to find the nonzero couplings from our fitting
results. For the L1-penalized MLE an estimated coupling is
considered as nonzero if | J est
ij | ⩾ z , where the z is an arbitrary
threshold. T o make use of the additional information of
uncertainty , for the v ariational Bayes couplings are considered
to be nonzero if | J ij  q 1 | ⩾ z ! (  i ) ( jj ) . The classification of
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CHRISTIAN DONNER AND MANFRED OPPER PHYSICAL REVIEW E 96 , 062104 (2017)
0 .. 10 50 . 9
p spar se
0 . 7
0 . 8
0 . 9
AU C
10 1 10 2 10 3
T
0 . 6
0 . 7
0 . 8
0 . 9
1 . 0
AU C
(a ( )b )
FIG. 3. The classification of nonzero couplings depending on
sparsity of couplings and data length. (a) The A UC depending on
the sparsity , i.e., the probability of a true coupling being 0, and in
(b) depending on length of training data T . Results for EM shown by
the red solid and the v ariational B ayes algorithm by the blue dashed
line. If not changed, parameters are as in Fig. 2 .
nonzero couplings is quantified by plotting the false positi ve
rate (proportion of zero couplings that are misclassified as
nonzero) versus the true positi ve rate (proportion of zero
couplings that are correctly classified as nonzero) for a
v arying threshold z ∈ [0 , ∞ ]. This is the Receiv er-Operator
characteristic (R OC) curve [see Fig. 2(c) for λ  and λ 
Bayes
respecti vely]. As a measure of classification performance we
use the area under the R OC curve (A UC), which is 1 for
perfect classification and 1 / 2 at chance lev el. Figure 2(d)
sho ws that performance for the EM and the variational Bayes
algorithm dif fer only marginally . For both algorithms the
A UC is approximately constant, when repeating the same data
generating and fitting procedure as before, but with v arying
sparsity p sparse [Fig. 3(a) ]. When increasing the length of
training data T the A UC increases as expected [Fig. 3(b) ]. For
subsequent analysis we will focus on the v ariational Bayes
algorithm.
C. Inference of biophysical netw ork
As an application of our algorithm we fit our model to data
generated from a more biologically plausible network. W e
simulate a recurrent network of 1000 leak y integrate-and-fire
neurons (800 excitatory and 200 inhibitory neurons) recei ving
Poisson input (see Appendix E and Ref. [ 30 ]). The synapses
connect neurons randomly , are conductance-based, and vary in
strengths and delays. The network is simulated for T = 1000 s.
Spike times of 30 e xcitatory and 10 inhibitory neurons are
used for fitting the kinetic Ising model, where neuron i is
considered as “acti v e” for 10 ms ( = γ − 1 ) after each spike
s i ( t ) = 1 and “inacti ve” otherwise [ s i ( t ) =− 1]. The two
questions we address here are (1) ho w well does the fitted
model reproduce the statistics of the recorded data and (2) ho w
are the synapses reflected in the estimated coupling parameters
J ?
For the first question we compare data obtained from the
spiking network with data sampled from the fitted the kinetic
Ising model  J  q 1 ( T = 1000 s). T o compare the original data
with the Ising model data the (second-order) correlations from
− 0 . 1 0 . 00 . 1
C data
ij
− 0 . 1
0 . 0
0 . 1
C model
ij
m iC
ij C ijk C ij kl
0 . 0
0 . 5
1 . 0
P earson Corr.
0 . 00 ..
F alse P ositiv e Rate
0 . 0
0 . 5
1 . 0
T rue Positiv e Rate
− 0 . 3 0 . 51 0 0 0 . 3
J q 1
− 0 . 3
0 . 0
0 . 3
J q 1
no synapse
synapse
(a) (b)
(c) (d)
FIG. 4. Model fitted to data from a simulated recurrent network.
(a) Second-order correlation C ij of the original data vs data
sampled from mean couplings  J  q 1 obtained via v ariational Bayes.
(b) Pearson correlation between first- and fourth-order correlations
of real sampled data. (c) R OC curve for identifying synapses with
posterior ov er couplings J (A UC = 0 . 65 ,λ 
Bayes = 16 . 5). (d) Mean
couplings  J  q 1 of the v ariational posterior vs transpose. Couplings
between neurons connected by a synapse are marked with gray
triangles.
these data are computed as
C ij = 1
T  T
0
[ s i ( t ) − m i ][ s j ( t ) − m j ] dt, (32)
where the mean is gi ven by m i =  T
0 s i ( t ) dt/T .
The results are compared in Fig. 4(a) , and we find good
agreement of original data and Ising samples. Furthermore,
we compute the higher order correlations C ij k and C ij k l of
the data and calculate the Pearson correlation coef ficient
between correlations from the original and the sampled data
[see Fig. 4(b) ]. The first two correlations ( m i and C ij ) yield
a Pearson correlation close to 1. Interestingly the Pearson
correlation coef ficient is strongly positiv e for C ij k and C ij k l as
well, indicating that the Ising model also carries information
about higher order correlations in the data.
As before we try to identify synapses in the simulated
network by R OC curve analysis [Fig. 4(c) ]. The classification
yields an A UC = 0 . 65 ( λ 
Bayes = 16 . 5). Even though there
is information about the synapses, many more nonzero
couplings are estimated that do not directly reflect synapses
in the network. This is possibly caused by the fact that the
network is only partially observ ed and the kinetic Ising model
compensates for this part with more nonzero couplings.
Pre vious work has indicated for the kinetic Ising model in
discrete time [ 15 , 31 ], that for experimental data recorded in
vivo the estimated couplings J sho w a symmetric signature:
J ij ≈ J ji . This is particularly interesting for the Ising model in
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continuous time, since for the model with symmetric couplings
the stationary distrib ution is gi ven by the maximum entropy
equilibrium model [ 14 ] and potentially justifies the use of static
Ising models for such data. As an indicator for symmetry
we plot the mean of the v ariational posterior obtained from
the recurrent network v ersus its transpose [Fig. 4(d) ]. W e
observe that man y couplings are indeed close to the diagonal,
while some sho w large de viations from it. Howe ver , those with
strong de viations correspond to the couplings which reflect
synapses in the underlying network. Hence, the approximately
symmetric part is not caused by synapses, but either by our
data transformation to fit the Ising model or by the fact that we
only partially observe the netw ork.
VII. DISCUSSION
In this paper we ha ve presented ef ficient algorithms for
inferring the couplings of a continuous time kinetic Ising
model defined by Glauber dynamics. Using a combination
of two auxiliary latent v ariable sets the complete data log-
likelihood becomes a simple quadratic function in the cou-
plings. A third set of auxiliary v ariables allows us to deal with
sparse couplings, equi v alent to an L1-penalized likelihood
without resorting to gradient-based algorithms [ 25 ]. Using
this representation we deri ve an EM algorithm for (penalized)
maximum likelihood estimation of the couplings with e xplicit
analytical updates. This leads to a guaranteed increase of the
likelihood in each iteration. The computational comple xity
is similar to a Ne wton-Raphson method for optimizing the
original log-likelihood, since the Hessian matrix requires a
similar in verse of the summed data co variances [ 15 ]. Ho we ver ,
our algorithm does not require any tuning of a step size.
W e hav e e xtended our latent v ariable approach to a
Bayesian scenario but ha ve restricted ourselv es to a fast
v ariational Bayes approximation. Howe ver , it is straightfor-
ward to de velop a Monte Carlo Gibbs sampler for the latent
v ariable structure. This would require drawing samples from
Pólya-Gamma density rather than computing only its mean.
W e hav e tested our inference algorithms on simulated data
demonstrating fast con vergence of the method. The v ariational
Bayes approximation allo ws us to perform model selection,
yielding hyperparameters which achie ve close to optimal like-
lihoods on test data. As an application of our approach we hav e
in vestig ated the quality of the kinetic Ising model to describe
data which were generated from a more realistic, biologically
inspired inte grate and fir e neural network model which is only
partially observed. W e ha v e sho wn that the kinetic Ising model
reproduces lo w order statistics of the data well. Ho we ver , the
partial observ ation of neurons prohibits a safe identification of
synapses in terms of the Ising coupling parameters. It would be
interesting to see if the performance of a kinetic Ising model on
such data could be improv ed by including explicit unobserved
neurons and their couplings in the model [ 32 ]. W e expect
that our latent v ariable approach would facilitate statistical
inference for such an extended model and pro vide alternativ es
to current approximate inference methods [ 33 – 36 ]. W e are
currently working on an e xtension of our inference approach by
including time-dependent model parameters which makes the
model more realistic and which has been sho wn of importance
for biological data analysis [ 12 , 37 ].
Finally , our latent v ariable approach should also be appli-
cable to other inference problems for point process models;
e.g., a combination with Gaussian process priors should allow
for nonparametric approximate inference of rate functions
for inhomogeneous Poisson processes [ 17 ]. Models with
similar point-process likelihoods are common in neuroscience
[ 38 – 40 ], for modeling seismic acti vity [ 41 ], analyzing social
network analysis [ 42 ], etc.
A CKNO WLEDGMENT
C.D. was supported by the Deutsche F orschungsgemein-
schaft (GRK1589/2).
APPENDIX A: GENERA TING D A T A
T o generate artificial data for the kinetic Ising model in
continuous time we can make use of the Gillespie algorithm
[ 16 ]. Ha ving a coupling matrix J for N spins and an initial
data vector s (0) data are generated as follows: (1) W e draw the
next update time t  from a e xponential distribution with mean
( γ × N ) − 1 , (2) we draw a spin i with probability 1 /N , and
finally (3) we flip spin i at time t  according to Eq. ( 2 ) and set
t ← t  . These three steps are repeated until t ⩾ T .
APPENDIX B: PROPER TIES OF PÓL Y A-GAMMA
DISTRIBUTION
The Pólya-Gamma density [ 18 ] allo ws us to represent the
in verse h yperbolic cosine function as an infinite Gaussian
mixture
cosh − b ( c/ 2) =  ∞
0
dω exp  − c 2
2 ω  p PG ( ω | b, 0) . (B1)
Furthermore, we define the tilted Pólya-Gamma distribution
as
p PG ( ω | b,c ) ∝ e − c 2 / 2 ω p PG ( ω | b, 0) . (B2)
From Eqs. ( B1 ) and ( B2 ) we obtain the moment generating
function
 e ωt = cosh b ( c/ 2)
cosh b  " c 2 / 2 − t
2 
. (B3)
By dif ferentiating ( B3 )a t t = 0 the analytical form of the
expectation of ω is obtained:
 ω = b
2 c tanh  c
2  . (B4)
APPENDIX C: LA TENT V ARIABLE REPRESENT A TION OF
LAPLA CE DISTRIBUTION
The Laplace distrib ution can written as an infinite mixture
of Gaussians [ 26 , 27 ]
λ
2 exp ( − λ | x | ) =  ∞
0  βλ 2
2 π exp  − βλ 2
2 x 2  p ( β ) dβ , (C1)
with
p ( β ) = ( β/ 2) − 2 ex p  − 1
2 β  . (C2)
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By inspection we find the conditional density
p ( β | x ) = p GIG ( β | x 2 λ 2 , 1 , − 1 / 2) , (C3)
where p GIG is a generalized in verse Gaussian distrib ution
defined as
p GIG ( β | a, b, ν ) = ( a/ b ) ν/ 2
2 K ν ( √ ab ) β ν − 1 ex p [ − ( aβ − b/ β ) / 2 ] ,
(C4)
and K ν is the modified Bessel function of the second kind. The
expectations of β are
 β = K 1 / 2 ( √ x 2 λ 2 )
√ x 2 λ 2 K − 1 / 2 ( √ x 2 λ 2 ) = 1
√ x 2 λ 2 , (C5)
where the Bessel functions cancel due to K ν ( √ x 2 λ 2 ) =
K − ν ( √ x 2 λ 2 ).
APPENDIX D: V ARIA TION AL BA YES
In the v ariational Bayes algorithm the updates in the step
updating q 2 in v olve the e xpectations  H t, n
i  q 1 and  ( H t, n
i ) 2  q 1
instead of only the pointwise MLE in the E step of the EM
algorithm. The required expectations are
 ω i ( t ) = 1
4 !  ( H i ( t )) 2  tanh[ !  [ H i ( t )] 2  ] ,
 ω n
i  =  ρ n
i 
4 "  H n
i  2 
tanh  "  H n
i  2  ,
 ρ n
i  = ( t n + 1 − t n ) γ exp  s n
i  H n
i 
2 cosh  "  H n
i  2 
.
(D1)
The free ener gy ( 27 ) that is minimized in the v ariational
Bayes algorithm is easy to calculate since we immediately
see that the terms in v olving p PG ( ω i ( t ) | 1 , 0), p PG ( ω n
i ) | ρ n
i , 0),
P Po ( ρ n
i | γ ( t n + 1 − t n )) and p ( β ) appear in the nominator as well
as in the denominator and cancel out. The free energy at a
minimum is
F ( q  ; p ) = 
( i,t ) ∈ F
ln 2 cosh[ !  [ H i ( t )] 2  ]
exp [ − s i ( t )  H i ( t )  ]
+ 
i,n
γ ( t n + 1 − t n ) ⎧
⎨
⎩
1 − exp  s n
i  H n
i 
2 cosh  "  H n
i  2 
⎫
⎬
⎭
+ 
i,j
ln ⎡
⎣
√ 2 π  J 2
ij  − 1 / 4
(2 √ λ ) 3 K − 1 / 2  " λ 2  J 2
ij  ⎤
⎦
− 
i  ln N  θ i   μ θ ,λ − 2
θ  + ln q 1 ( J )  , (D2)
where all expectations are tak en o ver the v ariational posterior
q  . Note the similarity of the first two summands and the
likelihood ( 4 ).
APPENDIX E: SIMULA TED NETWORK
OF SPIKING NEUR ONS
W e simulate a spiking network similar to the one described
in Ref. [ 30 ], Fig. 3 . The network consisted of three recurrently
connected populations of neurons: 800 input ( X ) neurons,
800 excitatory ( E ), and 200 inhibitory ( I ) neurons. The input
neurons do not get any input and generate Poisson spik es
independently with a rate of 10 Hz. For the conductance-based
integrate-and-fire neuron i in the population α ∈{ E, I } the
dynamics of the membrane potential V α
i are described by the
dif ferential equation
C m
dV α
i
dt =− g L  V α
i − V L  + 
β ∈{ X,E ,I }
I α,β
i ( t ) , if V α
i <V
th ,
(E1)
where the membrane capacitance is set to C m = 0 . 25 nF and
the leak conductance g L = 16 . 7 nS. The resting potential is
V L =− 70 mV, and the firing threshold V th =− 50 mV. After
each spike the membrane potential w as reset to V R =− 60 mV.
E and I neurons ha ve a 2 and 1 ms refractory period,
respecti vely . I αβ
i is the input current neuron i recei ves from
population β .
The neurons are connected with probability p connect =
0 . 2, and the connections consist of conductance-based
synapses (for details see the Supplementary Material
of Ref. [ 30 ]). W e draw the conductances for the
synapses from a uniform distrib ution with mean g αβ
and standard de viation 0 . 5 g αβ .A si nR e f .[ 30 ]w es e t
g EE = 2 . 4n S , g EI = 40 nS, g IE = 4 . 8n S , g II = 40 nS, and
g EX = g IX = 5 . 4n S .
For generating data we simulated the netw ork for T =
1000 s and recorded the spike times of a randomly selected
subpopulation (100 excitatory and 40 inhibitory neurons).
From those, the 30 excitatory and 10 inhibitory neurons with
the highest firing rates are selected as data for fitting the kinetic
Ising model.
T o preprocess the data for the Ising model we follow the
argument of Ref. [ 15 ]. The update rate γ can be interpreted as
the in verse of the width of a neuron’ s autocorrelation function,
which is typically found to be 10 ms. Hence we set γ = 10 2 Hz
and consider a neuron as “acti ve” for 10 ms after each
spike.
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Chapter 5. Inverse Ising pr oblem in c ontinuous time: A latent variable appr o ach
68

Chapter 6
Conjugacy b y augmen tation:
A dditional mo dels & p oten tial
extensions
In the previous c hapters w e ha v e demonstrated that the augmen tation sc heme describ ed in c hapter 2
is applicable to 3 differen t mo dels, and that the resulting inference algorithms are m uch faster than
previously prop osed ones.
As already men tioned the augmentation sc heme deriv ed in c hapter 2 is equiv alen t to com bining
the P ólya–Gamma (P olson et al. 2013) and P oisson pro cess augmen tation (Adams et al. 2009) .
Indep enden tly of the presen ted w ork the same augmen tation sc heme has b een utilised b y Lindon
(2018) , who implemen ted an efficient Gibbs sampler for one–dimensional problems of the sigmoidal
Gaussian Co x pro cess addressed in c hapter 3. Another recent w ork (Gonçalv es and Gamerman
2018) c hose a scaled Gaussian cum ulativ e densit y function as link function, whic h allo ws for Gibbs
sampling only using the P oisson pro cess augmen tation discussed in c hapter 2. Ho w ev er, sampling
from the conditional p osterior of the GP is not straigh tforw ard.
W e w ould like to emphasise, that the mo dels discussed in c hapters 3–5 are only an exemplary subset
of p ossible applications. F or example, a natural extension of Poisson processes are self–exciting
p oin t pro cesses, also kno wn as Ha wk es’ pro cesses (Ha wk es 1971) . Preceding ev en ts of such a
pro cess increase the lik eliho o d of follo wing even ts and are of in terest in mo delling of financial
mark ets (Em brec h ts et al. 2011) , seismic activit y (Ogata 1998) and neuronal data (Linderman and
A dams 2015) . So called line ar Ha wk es’ pro cesses ha v e the lik eliho o d as in Eq (2), c hapter 2, with
an in tensity function
Λ Z ( t ) = λ 0 + ∑
t ′ ∈H t
f ( t ′ ) ϕ ( t − t ′ ) ,
where
λ 0 ∈ R +
is the baseline in tensit y , and
H t
is the set of past ev en ts at time
t
.
ϕ
is a non–negativ e
memory k ernel and
f
(
t
) : [0
, T
)
→ R +
the time–dep enden t excitation amplitude. By c ho osing
f
(
t
) =
cσ
(
g
(
t
)) b eing a scaled sigmoid and
g
(
t
) a GP , w e can again utilise the augmentations
dev elop ed previously . F or non–line ar Ha wkes processes the intensit y is giv en b y
Λ Z ( t ) = f ( λ 0 + ∑
t ′ ∈H t
g ( t ′ ) ϕ ( t − t ′ ) ) ,
where
λ 0
and
g
(
t
) are not required to b e p ositiv e an y more, as long as
f
:
R → R +
. Surprisingly ,
when c ho osing
f
(
·
) =
cσ
(
·
) and
g
b eing a GP , the inference problem is ev en simpler than for the
69

Chapter 6. Conjugacy b y augmen tation: A dditional mo dels & p oten tial extensions
linear case and is closely related to the inference of the kinetic Ising mo del discussed in c hapter 5.
A dditionally , for a sp ecific likelihoo d for m ulti–class GP classification w e ac hiev e conjugacy of the
mo del with the 3 augmen tation schemata used in c hapter 4. F or details see app endix A, part I I I.
F urthermore, heteroscedastic GP regression problems (Lázaro-Gredilla and Titsias 2011) can b e
addressed, where the v ariance is space–dep enden t and its inv erse is mo delled b y the scaled sigmoid
ha ving a GP as argumen t. W e exp ect that the metho ds dev elop ed in this w ork can b e used to
p erform inference on man y additional mo dels.
A dmittedly , introducing the sigmoid link function in these mo dels can b e regarded as artificial
in order to utilise the presen ted augmentations. As discussed in c hapter 3 other link functions,
e.g. exp onen tial or squared, do not require (neither allo w for) a similar augmentation sc heme.
Ho wev er, they come with other disadv an tages. F or the v ariational approach with the exponential
link function the appro ximate p osterior’s v ariance is uncoupled from the data (Llo yd et al. 2014) .
Mo dels with the squared link function are limited to certain priors and domains. F urthermore, w e
exp erienced that inference results for mo dels with the squared link function dep end on ho w the
domain
X
is scaled. F or stabilit y of the algorithm from Llo yd et al. (2014) this scaling had to b e
adjusted. In contrast, w e did not exp erience suc h effects for the scaled sigmoid link function.
The P oisson pro cess augmen tation in Eq
(7)
, c hapter 2, can b e in v oked for all b ounded link
functions. How ev er, the prop ert y of the sigmoid function
σ
(
z
)=1
− σ
(
− z
) made the subsequen t
P óly a-Gamma represen tation p ossible. In general, even if one has a Gaussian represen tation of the
link function one requires in addition suc h a representation for the ‘complemen t’ function, whereas
for the sigmoid function w e can use the same representation.
In terestingly , a direct v ariational lo w er b ound has b een deriv ed for the sigmoid link function
without making use of P ólya–Gamma augmen tation (Jaakk ola and Jordan 2000; Mac k a y and
Gibbs 2000) . This b ound is equiv alen t to the v ariational one obtained with the Póly a–Gamma
augmen tation (W enze l et al. 2018) . P almer et al. (2006) discusses a class of functions for whic h
similar v ariational lo w er b ounds can b e deriv ed. This function class is broad and in teresting, b ecause
the deriv ation do es not require kno wledge (or ev en existence) of the augmen tation densities, e.g.
the P óly a–Gamma densit y . With the hindsigh t of the presen t w ork w e can deriv e a lo w er b ound for
the lik eliho o d in Eq
(2)
, c hapter 2 without making use of the marked P oisson pro cess augmen tation
(see app endix B, part I I I). Ho w ever, also here one requires
σ
(
z
) = 1
− σ
(
− z
) and w e w ould not b e
able to deriv e the v ariational b ound without this prop ert y .
The sampling sc heme that was utilised in c hapter 4 yields fast con v erging Mark o v c hains, where
the samples quic kly b ecome uncorrelated. F or suc h sc hemes kno wing the augmen tation density
is essen tial. The Sc hönberg theorem (Ressel 1976) shows, that translation and rotation in v arian t
k ernels can b e written as scale Gaussian mixture mo dels and that the augmen tation density exists.
Finding link functions, that are comp osed of these k ernels (Mercer 1909) and hav e the prop ert y
f
(
x
) = 1
− f
(
− x
) w ould allow for an augmen tation sc heme describ ed in c hapter 2. Sampling the
required densities, ho w ev er, is still an issue.
W e demonstrated sup erior efficiency of the newly deriv ed v ariational mean–fie ld algorithms compared
to more traditional metho ds, that assume an appro ximate Gaussian p osterior and optimise the
parameters b y gradient methods (Hensman et al. 2015b) . Because the latter approac h directly
maximises the lo wer bound on the mo del evidence
p
(
D
) , it finds the (lo cally) optimal v ariational
Gaussian p osterior. As w e ha v e seen empirically in c hapter 3, the v ariational p osterior obtained
with the augmen ted mo del and the original mo del p erform similar in terms of test lik eliho o d,
indicating that the gap b et w een the t w o p osteriors is relativ ely small. Poten tial impro v ements of the
augmen ted v ariational p osterior could b e obtained b y p ertubativ e corrections (Opp er et al. 2015)
as w as done by W enzel et al. (2018) . Alternatively , one could use the here prop osed v ariational
mean–field metho ds to find a p osterior that is close to an optim um, and then use this p osterior to
initialise gradien t metho ds (Hensman et al. 2015b) in order to find the optimal v ariational Gaussian
p osterior.
70

As w e hav e sho wn in the previous c hapters, the v ariational algorithm only requires
∼
10 iterations
to con v erge. Despite empirical evidence w e kno w little of wh y it conv erges quic kly . F or eac h up date
of the Gaussian mo del parameters w e need to solv e a quadratic problem, whic h is conv ex. Previous
w ork addressed v ariational inference for non–conjugate mo dels as w ell b y either reform ulating the
problem in to lo cally con v ex subproblems (Khan et al. 2013) , or making use of ‘partial’ conjugacy of
mo dels (Khan and Lin 2017) . Similar to the results presen ted here the optimisation pro cedures
con verge relativ ely fast and in just a few iterative steps. Ho wev er, using these metho ds for p oin t–
pro cess mo dels without the augmen tations presen ted here is still challenging, because the double
in tegral, i.e. the exp ectation ov er a random pro cess and the in tegration o ver the observ ed domain,
has to b e computed.
71

P art I I
Inference of mo dels for
non-stationary spiking data
73

Chapter 7
Statistical mo delling of spiking data:
A brief in tro duction
The neurosciences are no exception to the no w ada ys trend of collecting increasingly larger and
complex datasets. While this is true for neuroscience at all levels (molecular, single neuron, net w ork
and b eha vioural lev el), the a v ailabilit y of massiv e recordings of neuronal activit y w as esp ecially
fa voured b y the parallel adv ance in electroph ysiology and optogenetic techniques (Ahrens et al.
2013; Einev oll et al. 2012) . Suc h data are recorded in-viv o under increasingly complex exp erimen tal
paradigms, to relate cellular activit y to the organism’s b eha viour (Ishiy ama and Brec h t 2016;
Stensola et al. 2015) . Ev en though these data are recorded in tigh tly con trolled exp erimen tal
settings, the animal is affected b y m ultiple factors (e.g. mo o d, atten tional state, in ternal dynamics
etc.) whic h are commonly unkno wn to the researc hers and thus hard to con trol. This p oses a
c hallenge for the analysis of these data, b ecause not accounting for those phenomena migh t result in
wrong conclusions. Hence, semi– or unsup ervised inference metho ds for data analysis are required,
whic h are capable to extract these unkno wns from the recorded data.
A ttempts to mo del spiking activit y dates bac k to Ho dgkin and Huxley (1952) , describing a single
neuron dynamics b y a set of ordinary differen tial equations. Since then a range of mo dels ha ve
b een prop osed, whic h v ary in their complexity and lev el of abstraction (Abb ott and Kepler 1990) .
F or statistical description of spik e train data, likelihoo d–based approac hes ha v e b een suggested, i.e.
mo dels for whic h a data lik eliho o d can b e deriv ed (Cunningham and Y u 2014; Pillo w et al. 2008;
Sc hneidman et al. 2006) . Even though (or because) undoubtedly ov ersimplifying the picture, these
mo dels ha v e attracted great in terest, since they allo w pro jecting recorded data in lo w dimensional
parameter spaces.
Despite the fact that mo dels of lik eliho o d based approac hes can b e con v eniently fitted to data,
limitations and w eaknesses of these mo dels need to b e tak en into accoun t. Mo delling alw ays requires
assumptions ab out the underlying system. If those are not c hosen carefully , outcomes might be
misleading. F or example not accoun ting for non–stationarit y can lead to spurious p ositiv e results
in correlation analysis (Bro dy 1999) . F urthermore, b ecause of their phenomenological nature,
mo dels can describ e statistical prop erties of observ ed data but often do not pro vide a satisfying
explanation for these. An example are higher order correlations observed in spik e train data and
quan tified b y suc h phenomenological mo dels (Ohiorhen uan et al. 2010) . It was sho wn that simple
statistical mo dels can repro duce suc h higher order correlations (Mac k e et al. 2011; Shimazaki et al.
2015) ; These mo dels, how ev er, do not provide an y mec hanistic explanation of the phenomenon.
Recen tly , Shomali et al. (2018) used a minimal mechanistic in tegrate–and–fire (I&F) mo del to
in v estigate the underlying mec hanisms of the exp erimen tal observ ations.
Considering mo dels, that accoun t for sp ecific prop erties of the data might prev en t misin terpretation
75

Chapter 7. Statistical mo delling of spiking data: A brief in tro duction
and pro vide straigh tforw ard exp erimen tal hypotheses. Ho w ev er, this usually comes with increasing
mo del complexit y , whic h in turn requires more complicated fitting pro cedures and more data.
Cho osing the ‘b est’ mo del for recorded spik e train data usually results in finding a compromise
b et w een those t w o extremes.
While analysing in–viv o data that are recorded from b eha ving animals non–stationarit y has to b e
accoun ted for. In fact, it has b een sho wn that for resting animals (T so dyks et al. 1999) and even
for neurons in cell–cultures (Sasaki et al. 2007) the stationary assumption can b e violated. In the
follo wing, we address t w o different models which accoun t for temp oral v ariabilit y of spike–train
data recorded from m ultiple neurons. These mo dels are based on t w o differen t lev els of abstraction,
where one is more con v en tional and the other one tries to minimise the gap b et w een mo del and the
biological system.
Outline of part I I
First, in c hapter 8 w e consider a non–stationary extension of the mo del presen ted in c hapter 5.
The mo del assumes that observ ed spik e–train data are generated b y a kinetic Ising mo del, a simple
dynamics for binary data considering couplings among neurons. T o accoun t for non–stationarit y ,
w e make the simplifying assumption that the model parametrisation (i.e. effectiv e couplings)
in termitten tly jumps in to differen t states. F or the time b etw een jumps, the parameters are constan t.
The n um b er of mo del’s states are finite. Previously describ ed augmen tation sc hemes combined with
v ariational inference (Donner and Opp er 2017) allo w for fitting this fully Bay esian mo del efficien tly
to spik e data. F urthermore, w e sho w that this metho d do es not require explicit binning of data,
an assumption whic h is required for the ma jorit y of lik eliho o d–based mo dels and migh t affect the
in terpretation of results.
F or the second mo del, presen ted in chapter 9, w e assume a minimal bioph ysically plausible spiking
mec hanism, namely the integrate-and-fire (I&F) class (Gerstner and Kistler 2002) . In teresting from
the inference p ersp ectiv e is that for this class of ph ysiologically constrained mo dels it is p ossible
to deriv e a likelihoo d semi analytically , b y solving the first passage time problem (Ladenbauer
et al. 2018; Mullo wney and Iyengar 2008) . These lik eliho o ds ha v e b een used relatively little so
far b ecause optimisation is though t of as b eing demanding compared to more phenomenological
mo dels. Ho w ev er, Laden bauer et al. (2018) show ed that inference for these mo dels is p ossible in
the stationary case. In this w ork, w e consider a non–stationary scenario, w here a population of
I&F neurons is driv en by a time–dependent stochastic input curren t. W e sho w, that for this mo del
a Hidden Mark ov model can b e deriv ed, for whic h the lik eliho o d can b e efficien tly ev aluated and
optimised.
76

Chapter 8
Unpublished article: Bayesian
network infer enc e fr om
non-stationary spiking data
Authors:
Christian Donner 1 , 2 , Manfred Opp er 1 , 2
1 T echnisc he Univ ersität Berlin. 2 Bernstein Cen ter for Computational Neuroscience Berlin.
Chapter 8
This c hapter comprises the unpublished manuscript, whic h is authored b y m yself
(CD), and Prof. Manfred Opp er (MO).
Con tributions :
CD and MO conceiv ed and designed the work. CD deriv ed the inference algorithms and dev elop ed
the Python co de. CD p erformed the numerical experiments. CD wrote the man uscript.
Python co de on GitHub: https://gith ub.com/c hristiando/MJP_ising_inference.git
77

Inference fr om non-st a tionar y spiking d a t a
Ba y esian net w ork inference from non-stationary spiking data
Christian Donner [email protected]
Manfred Opp er [email protected]
A rtificial Intel ligenc e Gr oup
T e chnische Universit¨ at Berlin
Berlin, Germany
Abstract
W e prop ose a mo del for non–stationary , correlated spiking data recorded from multiple
neurons in con tinuous time. The correlated activit y is mo delled b y a kinetic Ising mo del,
whic h considers effective couplings betw een the observ ed neurons. T o account for the non–
stationarit y in spiking data, the parametrisation of the Ising mo del is assumed to follo w a
Mark ov jump process, which can assume a finite n um b er of states. W e derive an efficien t
v ariational inference scheme for this model using a structured mean–field assumption. The
resulting unsup ervised algorithm accurately reco v ers the Mark ov jump process, the dynamic
coupling structure of the net work, and the probabilities of the Mark o v jump pro cess states.
Finally , we demonstrate practicalit y on m ulti-unit recordings from monkey V4. The mo del
reco vers the relev an t features of the b eha vioural task and additionally un v eils patterns of
activit y uncorrelated to the exp erimen tal paradigm.
Keyw ords: Non–stationary spiking data, Marko v jump pro cess, kinetic Ising mo del,
monk ey V4
1. In tro duction
T ec hnical adv ances in the field of extracellular recordings of ensem bles of neurons (Stev en-
son and Kording, 2011) require no v el and adequate analysis tec hniques to uncov er ho w
information is pro cessed in the cen tral nerv ous system. These techniques should accoun t
for the dynamic and correlated nature of these data (i.e. spike tr ains ). The frequently made
stationarit y assumption is strongly violated, when data are recorded in-viv o from b ehav-
ing animals. Ev en spik e trains recorded from an animal at rest, i.e. sp on taneous activit y
(Tso dyks et al., 1999), or recorded in–vitro (Sasaki et al., 2007) exhibit qualitativ e c hanges
o v er time. Appropriate statistical description of these data th us requires flexible mo dels.
This problem has b een addressed b y in tro ducing laten t hidden states, that follo w a
certain dynamics o ver the recording time. Differen t works consider either a con tin uous
dynamics of the laten t state (Y u et al., 2009; La whern et al., 2010; Zhao and P ark, 2017),
or piecewise constan t and in termitten tly jumping states (Ab eles et al., 1995; Escola et al.,
2011; Putzky et al., 2014). F or the latter type it is often additionally assumed, that latent
dynamics can revisit a hidden state m ultiple times (Ab eles et al., 1995; Putzky et al.,
2014; Escola et al., 2011; Stim b erg et al., 2012). These approac hes are closely related to
Diric hlet–pro cesses (T eh, 2011). Despite the simplification in tro duced b y these assumptions,
exp erimen tal studies sho w that the appro ximations are plausible in some scenarios (Sasaki
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Donner and Opper
et al., 2007; Latimer et al., 2015; P onzi and Wick ens, 2010). Ho w ev er, most of these w orks
consider discrete time and hence require the binning of data recorded in con tinuous time.
T o describ e neuronal activit y a p opular mo del class is generalised linear mo dels (GLMs)
for P oin t pro cesses (Pillo w et al., 2008; Lawhern et al., 2010; Gerwinn et al., 2010). These
mo dels assume that observ ed spiking activit y is a probabilistic pro cess. The rate of this pro-
cess is parametrised in v arious w ays and usually depends on observed factors, e.g. stim ulus,
activit y of other neurons, etc. Inference for GLMs is p erformed in sev eral wa ys to obtain
the maxim um a–p osteriori (MAP) estimate (Pillo w et al., 2008; T yrc ha and Marsili, 2013),
or a full p osterior of the mo del parameters (Gerwinn et al., 2010). The latter (Ba y esian) ap-
proac h requires some kind of appro ximation, b ecause the GLM lik eliho o ds are of a form, for
whic h the p osterior solution is in tractable. F or time–discretised data the GLM (Bernoulli)
lik eliho o d can b e rendered in a fa vourable form b y v ariable augmen tation (Polson et al.,
2013). This form allows to efficien tly sample the p osterior (Linderman et al., 2016), or to
obtain the MAP estimate b y a fast exp ectation–maximisation (EM) algorithm (Scott and
Pillo w, 2012; Scott and Sun, 2013). Ho wev er, in the con tin uous time limit this augmentation
is not practical an ymore, b ecause GLM lik eliho o ds in v olve an exponential whose argumen t
con tains an in tegral o v er time. F ortunately , recen t work pro vides extended augmen tation
sc hemes (Donner and Opp er, 2017, 2018) to deal with suc h lik eliho o ds.
A sp ecific case of a GLM in discrete time is the kinetic Ising mo del considering couplings
to other observ ed neurons. Prop osed in the 70’s (Little, 1974) to describ e computations of
the brain it w as recen tly utilised for statistical description of correlated spiking data (T yrc ha
and Marsili, 2013; Dunn et al., 2015; Marre et al., 2009). An in teresting extension of this
mo del is the kinetic Ising mo del with asynchr onous up dates (Zeng et al., 2013), whic h allo ws
to study the transition from discrete to con tin uous time. In the con tin uous time limit this
mo del b ecomes a Mark o v jump pro cess, where neurons can switch betw een on– and off–
states. In statistical ph ysics this mo del is kno wn for describing dynamics of interacting
spins, the so–called Glaub er dynamics (Glaub er, 1963). While the kinetic Ising mo del with
async hronous up dates do es not b elong to the p opular class of GLMs, it con tains as sub class
the equilibrium Ising mo del (Glaub er, 1963), whic h has gained atten tion in the neuroscience
comm unit y in recen t y ears (Sc hneidman et al., 2006; Shlens and Field, 2006; Mana et al.,
2018).
In this w ork w e com bine the kinetic Ising mo del with async hronous up dates with a
laten t state space dynamics in order to describ e non–stationary con tin uous time spiking
data of m ultiple neurons. First w e describ e the generativ e mo del for the discrete time
case and obtain a new fa vourable form of the model likelihoo d by v ariable augmen tation
(section 2). This allows us to deriv e an efficien t inference algorithm in section 3, that infers
the couplings of differen t latent states, the time p oin ts the system sp ends in eac h state,
and ho w lik ely are those states to app ear in a fully Ba y esian framew ork via v ariational
metho ds. In section 4, we consider the con tin uous time limit of the mo del and the resulting
inference algorithm. The mo del parameters can b e correctly reco v ered on an artificial
dataset. Finally , in section 5, we demonstrate practicalit y on a dataset of 40 m ulti– and
single unit activit y recorded in monkey V4.
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Inference fr om non-st a tionar y spiking d a t a
2. Data and Generativ e Mo del
Data Binary spik e train data s 0: T are represen ted in a ( T ∆ − 1 ) × N dimensional matrix,
where N is the n um b er of neurons observ ed, and T is the recording length discretised in
bins of width ∆. F urthermore, we define the set of time points as T = { ∆ , 2∆ , . . . , T } . A t
eac h time p oin t t ∈ T the observ ed data of the net w ork can b e represen ted b y a v ector
( s t, 1 , . . . , s t,N ) > , where s t,i = 1 if a neuron i is activ e in the in terv al t ∈ [ t, t + ∆) and − 1
otherwise.
Observ ation mo del A t eac h time p oin t t ∈ T the data are sampled from a kinetic
Ising mo del with async hronous up date Zeng et al. (2013) ha ving time-dep enden t external
fields θ ( t ) and coupling matrix J t . It can b e mo delled a doubly sto c hastic pro cess, where
at eac h time p oin t t the activit y of each neuron i is up date d with a probabilit y ∆ γ s . Only
if the neuron is up dated its state is flipp ed s t +∆ ,i = − s t,i with probabilit y
P flip
t,i = exp ( − s t,i H t,i )
2 cosh ( H t,i ) , (1)
where H t,i = θ t,i + P N
j =1 J t,ij s t,j . θ t,i denotes the external field of neuron i and J t,ij is the
coupling from neuron j to neuron i at time t . P and Q denote probabilities throughout
this w ork, while p and q are densities with resp ectiv e measures P and Q . F or notational
con v enience the v ector form H t,i = J >
t,i s t is in tro duced, where J t,i = ( J t,i 0 , . . . , J t,iN ) > ,
and s t = ( s t, 0 , . . . , s t,N ) > . In this notation the external fields are J t,i 0 = θ t,i and the data
comprise a hidden neuron whic h is alw a ys activ e denoted b y s t, 0 = 1 for t ∈ { 0 }∪T .
F urthermore, w e define J t = ( J t, 1 , . . . , J t,N ).
Time dep endence of the mo del parameters W e consider a sequence of laten t states
z 0: T = ( z 0 , . . . , z T ), where z t ∈ 1 , . . . , K . Eac h latent state k has a corresp onding coupling
matrix J k . If at time t the latent state z t = k , then J t = J k . Hence, w e define the function
H t,i ( z ) = P K
k =1 δ z t ,k J >
k ,i s t and P flip
t,i ( z ) according to (1). δ x,y is the Kronec k er delta. The
lik eliho o d of the data giv en the couplings of all states J 1: K and the laten t state sequence
z 0: T is
P ( s 0: T | J 1: K , z 0: T ) = Y
( t,i ) ∈F
∆ γ s P flip
t,i ( z ) Y
( t,i ) ∈N F  1 − ∆ tγ s P flip
t,i ( z )  , (2)
where F and N F are the sets of ( t, i ) pairs of flips and non flips, resp ectiv ely . F urthermore,
w e define the sets F i and N F i con taining only the (non) flip times of neuron i .
Priors The prior dynamics o v er the state v ariables is assumed to b e a Mark ov c hain.
A t eac h time p oin t the laten t state might switc h with probabilit y ∆ γ z , where γ z is the
state switching r ate . If the laten t state switc hes, the new state is chosen out of K states
according to a m ultinomial state distribution P ( z t +∆ = k | state switc h at t ) = π k . Hence,
the transition probabilit y of the Marko v c hain is
P ( z | z 0 , π 1: K ) = ( ∆ γ z π z if z 6 = z 0
1 − ∆ γ z (1 − π z ) ≈ exp ( − ∆ γ z (1 − π z )) else ,
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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Donner and Opper
where π 1: K = ( π 1 , . . . , π K ) > and the appro ximation holds for ∆ γ z << 1 b eing exact in the
limit of ∆ → 0. The probabilit y of a sequence z 0: T is
P ( z 0: T | γ z , π 1: K ) = Y
t ∈T
P 0 ( z t | z t − ∆ , π 1: K ) P ( z 0 ) . (3)
W e define P 0 ( z 0 = k ) = K − 1 as the initial uniform state distribution at t = 0.
Because w e exp ect sparse coupling structures in neuronal net w orks, w e assume a Laplace
prior for eac h of the couplings J k ,ij for j = 1 , . . . , N and k = 1 , . . . , K with mean µ J = 0
and scaling parameter σ J / √ N . The external fields θ k ,i ha ve a Gaussian prior with mean
µ θ and v ariance σ θ . The state probabilities π 1: K are distributed according to a Dirichlet
prior with concen tration parameters α /K for all states k = 1 , . . . , K . W e a v oid dealing with
infinite Diric hlet–pro cess priors b y assuming a finite, but large enough K . F urthermore, w e
assume an exp onen tial distribution as prior with mean 1 (whic h means if ∆ is in units of
seconds, w e exp ect 1 state switc h p er second) o v er the switc hing rate γ z .
Data augmen tation Inference for the prop osed model is difficult b ecause of the non–
conjugacy of lik eliho o d Eq (2) to the priors. T o render the lik eliho o d and the prior into an
alternate, conjugate Gaussian form w e utilise the augmentation sc heme dev elop ed b y Don-
ner and Opp er (2017). Along these lines we mak e use of the w ell-kno wn P´ olya–Gamma
represen tation (P olson et al., 2013) and rewrite the flip probabilit y (1)
P flip
t,i ( z ) = Z ∞
0
exp  − s t,i H t,i ( z ) − 2( H t,i ( z )) 2 ω t,i − ln 2  p ( ω t,i ) dω t,i .
F or notational con venience, w e define
p flip
t,i ( z , ω ) .
= exp  − s t,i H t,i ( z ) − 2( H t,i ( z )) 2 ω t,i − ln 2  p ( ω t,i ) .
W e write the factors of the second pro duct in Eq (2) as an exp ectation o v er a binary v ariable
ρ t,i
(1 − ∆ γ s P flip
t,i ( z )) = (1 − ∆ γ s + ∆ γ s P noflip
t,i ( z )) = X
ρ t i ∈{ 0 , 1 }  P noflip
t,i ( z )  ρ i,t Ber( ρ i,t | ∆ γ s ) ,
where Ber( ρ | ∆ γ ) denotes a Bernoulli distribution o ver random v ariable ρ = 1 with proba-
bilit y ∆ γ . F urthermore, w e ha ve
P noflip
t,i ( z ) = 1 − P flip
t,i ( z ) = exp ( s t,i H t,i ( z ))
2 cosh ( H t,i ( z )) .
By applying once more the P´ oly a–Gamma augmen tation w e get
(1 − ∆ γ s P flip
t,i ( z )) = X
ρ t i ∈{ 0 , 1 } Z R +
p noflip
t,i ( z , ρ, ω ) dω ,
where
p noflip
t,i ( z , ρ, ω ) =  exp  s t,i H t,i ( z ) − 2( H t,i ( z )) 2 ω t,i − ln 2  p ( ω t,i )  ρ t,i Ber( ρ t,i | ∆ γ s ) .
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Inference fr om non-st a tionar y spiking d a t a
Figure 1: Graphical mo del for data at time t. Grey circle denotes observ e d data, and
white circles r andom v ariables. Hyp erparameters, data, and random v ariables at
the neigh b ouring time–ste p s are depicted w i thout circles.
This allo ws us to rewrite the join t augmen ted lik eliho o d of Eq 2 as
p ( s 0: T , ω F , ( ρ, ω ) N F | J 1: K , z 0: T ) = Y
( t,i ) ∈F
∆ γ s p flip
t,i ( z , ω ) Y
( t,i ) ∈N F
p noflip
t,i ( z , ρ, ω ) , (4)
where ω F and ( ρ, ω ) N F are the sets of augmen tation v ariables at the flip and non flip times,
resp ectiv ely .
F or the prior o v er the couplings J k ,ij w e in v ok e the fact that the Laplace densit y can b e
rewritten as a scale Gauss i an mixture mo del (Gao, 2008)
p ( J k ,ij ) = 1
2 σ J
exp  − | J k ,ij |
σ J  = Z s β k ,ij
2 π σ 2
J
exp − β k ,ij ( J k ,ij ) 2
2 σ 2
J ! η ( β k ,ij ) dβ k ,ij ,
where η ( β ij ) is an inverse gamma distribution with shap e parameter b eing 1 an d scale
parameter 1
2 . The new set of v ariables β is named sp arsity variables , and w e define β k
b eing the sparsit y v ariables for J k .
Join t distribution With the augmen tation w e ha v e a mo del, that has a G au s sian form
in terms of the couplings J 1: K . The join t distribution of the mo del is
p ( s 0: T , ω F , ( ρ, ω ) N F , β 1: K , z 0: T , γ z , π 1: K , J 1: K | ϑ ) = p ( s 0: T , ω F , ( ρ, ω ) N F | J 1: K , z 0: T )
× P ( z 0: T | π 1: K , γ z ) p ( π 1: K | α )
× p ( J 1: K , β 1: K | µ θ , σ θ , µ J , σ J ) p ( γ z ) ,
(5)
where the set of h yp erparameters is denot e d b y ϑ .
= { µ θ , σ θ , γ s , α , µ J , σ J } . Note, that b y
marginalising o v er the sets of augmen te d v ariables ω F , ( ρ, ω ) N F , β 1: K one obtain s the join t
distribution of the original mo del again. A graphical represen tation of the full mo del can
b e seen in Fig 1.
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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[ms]

Figure 2: Data from generativ e mo del. A The kinetic Ising mo del par am etrisation s of
the 4 most lik ely states, that generate the data. The v ector on the left denote the
external fields θ k and the matrices the couplings J k . The frame colour indicates
the state iden tit y . B State distribution sampled from a stic k-breaking pro c ess
with α = 1. C P erio ds of the generated data, where eac h ro w is the activit y of
one neuron. Blac k indicates a neuron b eing in th e activ e state . C ol ours indicate
differen t s tates the data at a giv en time are sam p le d from.
Con tin uous time limit It is p ossible to obtain the con tin uous time limit ∆ → 0 for the
original and the augmen ted mo del i n Eq (5). The prior of se qu e n c e z 0: T in Eq (3), will
b ecome a Mar k o v jump pro cess densit y . The limit of the ki netic Ising mo del lik eliho o d (2)
and its augmen ted coun terpart in Eq (4) has b een already deriv ed b yDonner and Opp er
(2017). W e refrain from taking the limit at this p oin t and rather will deriv e an inference
algorithm for the discrete time mo del, whic h pro vides effi cien t iterativ e up dates for an
appro ximate p osterior. In the end w e sho w that for eac h up date the con tin uous tim e limit
exists.
3. V ariational Inference
While v ariou s efficien t inference algor ithms could b e deriv ed for the augmen ted mo del (see
section 6), w e resort to a h yp othetical fast and fully Ba y esian, but appro ximate v ariational
approac h. F or solving the inference problem w e mak e the structured mean–field assumption
of the form
p ( ω F , ( ρ, ω ) N F , β 1: K , z 0: T , γ z , π 1: K , J 1: K | s 0: T , ϑ ) ≈ q ( ω F , ( ρ, ω ) N F , β 1: K , z 0: T , γ z , π 1: K , J 1: K )
.
= q 1 ( ω F , ( ρ, ω ) N F , β 1: K , z 0: T )
× q 2 ( J 1: K , π 1: K ) q 3 ( γ z ) .
(6)
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Inference fr om non-st a tionar y spiking d a t a
W e iden tify the variational lower b ound on the logarithm of the marginal likelihoo d as
L ( q ) = E Q  ln p ( s 0: T , ω F , ( ρ, ω ) N F , z 0: T , γ z , π 1: K , J 1: K , β 1: K | ϑ )
q ( ω F , ( ρ, ω ) N F , β 1: K , z 0: T , γ z , π 1: K , J 1: K )  ≤ ln p ( s 0: T | ϑ ) ,
where E Q [ · ] is the exp ected v alue with resp ect to the corresp onding v ariational p osterior
measure of the densit y in Eq (6). W e mak e use of the fact, that the optimal factors of a
mean–field p osterior Q x q x ( Y x ) are giv en b y
ln q x ( Y x ) = E Q \ x  ln p ( s 0: T , Y x , Y \ x | ϑ )  + const . , (8)
where the exp ectation is with resp ect to the p osterior measure o v er all v ariables Y \ x except
Y x . W e deriv e up dates for eac h of the three v ariational factors in Eq (6) given the other
t w o factors.
3.1 First factor
F rom the join t likelihoo d in Eq (5), the v ariational p osterior in Eq (6), and Eq (8) we deriv e
the factorisation
q 1 ( ω F , ( ρ, ω ) N F , β 1: K , z 0: T ) = q ( ω F | z 0: T ) q (( ρ, ω ) F | z 0: T ) q ( z 0: T ) q 1 ( β 1: K ) ,
meaning that ω F and ( ρ, ω ) N F are conditionally indep enden t giv en z 0: T . The sparsit y
v ariables are indep enden t of the rest β 1: K .
Augmen ted v ariables at the flip times The conditional distribution turns out to b e
a pro duct of tilted P´ olya–Gamma densities
q 1 ( ω F | z 0: T ) = Y
( t,i ) ∈F
p PG ( ω t,i | 1 , c t,i ( z )) ∝ Y
( t,i ) ∈F
exp  − [ c t,i ( z )] 2
2 ω i,t  p PG ( ω t,i | 1 , 0) ,
where c t,i ( z ) = 2 q E Q \ 1 [( H t,i ( z )) 2 ]. This defines the conditional exp ectation
E Q [ ω i,t | z t ] = 1
2 c t,i ( z ) tanh  c t,i ( z )
2  .
F or future deriv ations w e define
ˆ p flip
t,i ( z ) =
exp  − s t,i E Q \ 1 [ H t,i ( z )] 
2 cosh  c t,i ( z )
2  .
Augmen ted v ariables at non-flips F or the conditional distribution of the ( ρ, ω ) t,i pairs
in N F w e obtain a pro duct of conditional P´ oly a–Gamma and Bernoulli distributions
q (( ρ, ω ) N F | z 0: T ) = Y
( t,i ) ∈N F
[ p PG ( ω t,i | 1 , c t,i ( z ))] ρ t,i Ber( ρ t,i | P ( ρ )
t,i ( z t )) , (9)
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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Donner and Opper
where
P ( ρ )
t,i ( z ) = ∆Λ t,i ( z )
(1 − ∆ γ s ) + ∆Λ t,i ( z ) with Λ t,i ( z ) = γ s
exp  s t,i E Q \ 1 [ H t,i ( z )] 
2 cosh  c t,i ( z )
2  .
A random pro cess as in Eq (9) can b e easily sampled, b y first sampling the ρ sequence and
then the v ariables ω t,i , where ρ t,i = 1. The exp ectation ov er a pair of v ariables is giv en by
E Q [ ω t,i ρ t,i | z t ] = E Q [ ω t,i | z t ] P ( ρ )
t,i ( z ) .
State lab els o v er time Next w e deriv e the v ariational p osterior o ver the sequence of
laten t states z 0: T . With Eq (8) and marginalising out ω F and ( ω , ρ ) N F , w e obtain
Q ( z 0: T ) ∝ Y
t ∈T
P 1 ( z t | z t +∆ ) exp ( − U t ( z t , z t − ∆ )) P ( z 0 ) , (10)
where w e defined a new effective transition matrix
P 1 ( z | z 0 , ∆) = δ z ,z 0 + ∆ ϕ ( z | z 0 ) ,
with effectiv e rates b eing
ln ϕ ( z | z 0 ) = E Q \ 1 [ln π k ] + E Q \ 1 [ln γ z ] , if z 6 = z 0 and
ϕ ( z | z ) = − X
z 0 6 = z
ϕ ( z 0 | z ) .
The U -function is giv en b y
U t ( z , z 0 ) = U data
t ( z ) + ˜
U ( z , z 0 ) ,
where the first term is the new effectiv e negative data log–lik eliho o d
U data
t ( z ) = −
N
X
i =1  1 F (( t, i ))
∆ ln ˆ p flip
t,i ( z )+Λ t,i ( z )  ,
with 1 X ( x ) b eing 1 if x ∈ X and 0 otherwise. The second term
˜
U ( z , z 0 ) = δ z ,z 0  1
∆ ln(1 + ∆ ϕ ( z | z 0 )) + E Q \ 1 [ γ z ] (1 − E Q \ 1 [ π z ])  .
is a discrepancy term of the probabilit y staying in the same state caused b y the mean–
field assumption in Eq (6). If the term w ould not in v olv e exp ectations it w ould v anish (b y
noting that ln(1 − ∆ ϕ ( z t +∆ | z t )) ≈ ∆ ϕ ( z t +∆ | z t ) for ∆ ϕ ( z t +∆ | z t ) << 1). With Eq (10) w e
see immediately , that the p osterior represen ts a Marko v c hain of the form
Q ( z 0: T ) = Y
t ∈T
Q t ( z t | z t − ∆ ) Q 0 ( z 0 ) , (11)
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Inference fr om non-st a tionar y spiking d a t a
where the factors ha v e to b e determined. F ollo wing standard pro cedures for hidden Marko v
mo dels (Bishop, 2006) or minimising the Kullbac k–Leibler div ergence (see app endix B) we
can deriv e the transition probabilities
Q t ( z | z 0 ) ∝ P 1 ( z | z 0 , ∆) exp  − ˜
U ( z , z 0 )∆  r t ( z ) .
The r t factors are so-called bac kward messaged, that can be solved recursiv ely b y
r t ( z ) = X
z 0
P 1 ( z 0 | z , ∆) exp  − U t +∆ ( z 0 , z )∆  r t +∆ ( z 0 ) , (12)
where w e initialise r T ( z ) = 1 for z = { 1 , . . . , K } . With these results we actually are able
to solv e for the factors of the Mark o v Chain in Eq (11), by first solving the bac kw ard
equations and then obtaining the marginals b y forw ard iterating Eq (11). Ho wev er, w e
w ould lik e to ha v e forw ard iterations b eing indep enden t of the backw ard messages, such
that w e can parallelise their computation. In order to do so, w e define the marginals as
Q t ( z ) .
= f t ( z ) × r t ( z ), where f t ( z ) are the forw ard messages. Consequen tly , w e can derive
iterativ e up dates
f t ( z ) = X
z 0
P 1 ( z | z 0 , ∆) exp  − U t ( z , z 0 )∆  f t − ∆ ( z 0 ) , (13)
whic h is indep enden t of r t and where f 0 ( z ) = P ( z 0 ).
Sparsit y v ariables Eac h sparsit y v ariable β k ,ij is indep enden t of all other laten t v ariables
in the v ariational p osterior densit y and hence can b e separately up dated. The appro ximate
p osterior densit y is gener alise d inverse Gaussian density
q ( β k ,ij ) = ( a k ,ij ) − 1 / 4
2 K − 1 / 2  √ a k ,ij  ( β k,ij ) − 3
2 exp  − a k ,ij β k,ij + 1 /β k ,ij
2  ,
where a k ,ij = E Q h ( J k,ij − µ J ) 2 i /σ 2
J , and K y ( x ) is the mo dified Bessel function. The ex-
p ectation o v er the sparsit y v ariables are E Q [ β k ,ij ] = ( a k ,ij ) − 1 / 2 .
3.2 Second factor
F or the second factor has a factorising form
q 2 ( J 1: K , π 1: K ) = q ( π 1: K )
K
Y
k =1
N
Y
i =1
q ( J k ,i ) ,
meaning that state lab els π 1: K are indep enden t of the couplings J 1: K . F urthermore, the
couplings J k ,i of eac h state k and neuron i are indep enden t of all other couplings.
Couplings Due to the augmen ted mo del and the mean field assumption w e are able to
deriv e the p osterior densit y for eac h v ector J k ,i =  J k
i 0 , . . . , J k
iN  > in a closed form. W e
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iden tify them b eing Gaussian densities with co v ariance matrix
Σ k ,i = 4 X
t ∈T h 1 F i ( t ) E Q \ 2 [ δ z t ,k ω t,i ] + 1 N F i ( t ) E Q \ 2 [ δ z t ,k ρ t,i ω t,i ] i C t − ∆ +  ˜
Σ k ,i  − 1 ! − 1
,
(14)
where w e defined the data cov ariance matrix C t = s t s >
t and ˜
Σ k ,i is a diagonal matrix with
diag( ˜
Σ k ,i ) =  σ 2
θ , ( σ J ) 2
E Q [ β k,i 1 ] , . . . , ( σ J ) 2
E Q [ β k,iN ]  . F or the mean we get
µ k ,i = Σ k
i X
t ∈T h − 1 F i ( t ) E Q \ 2 [ δ z t ,k ] + 1 N F i ( t ) E Q \ 2 [ δ z t ,k ρ t,i ] i C t − ∆ ,i +  ˜
Σ k ,i  − 1 µ J ! ,
(15)
with C t,i = s t,i s t . Note, that those equations are analogue to the results obtained for
the v ariational p osterior b y Donner and Opp er (2017). They differ only b ecause Eq (14)
and (15) in v olv e exp ectations of δ z t ,k , whic h act as ‘w eigh ting factors’ for states k at time
p oin t t .
State probabilities The v ariational p osterior densit y of state probabilities cannot b e
obtained in close form. F or this reason w e further appro ximate q ( π 1: K ) with a Diric hlet
distribution ha ving concen tration parameters α = ( α 1 , . . . , α K ) > . W e maximise the v ari-
ational lo w er b ound (Eq 7) b y differen tiating it with resp ect to parameters α and up date
them using a simple gr adient asc ent algorithm. Once con v ergence is ac hiev ed the distribu-
tion of state probabilities is giv en by
q ( π 1: K ) =
Γ  P K
k =1 α k 
Q K
k =1 Γ( α k )
K
Y
k =1
π α k − 1
k ,
where Γ( x ) is the gamma function.
3.3 Third factor
Switc hing rate F or the switc hing rate γ z the v ariational p osterior densit y is a gamma
densit y
q 3 ( γ z ) = ν κ
Γ( κ ) γ κ − 1
z exp ( − ν γ z ) , (16)
with shap e parameter and scale parameters
κ = X
t ∈T X
z
(1 − Q t ( z | z )) Q t − ∆ ( z )+1
ν = X
t ∈T X
z
Q t ( z | z ) Q t − ∆ ( z )(1 − E Q [ π z ])∆ + 1 .
V ariational lo w er b ound Since the assumed p osterior densit y q is no w completely
kno wn and all required exp ectations can b e deriv ed analytically , it is straightfor w ard to
ev aluate the v ariational lo w er b ound in Eq 7 to c hec k con vergence after eac h up date of the
v ariational p osterior.
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Inference fr om non-st a tionar y spiking d a t a
4. The con tin uous time limit
W e no w study the limit ∆ → 0 of the up dates for the v ariational p osterior densit y derived
in section 3.
First factor The p osterior densities of ω F and the sparsit y v ariables β 1: K do not dep end
on ∆ and consequen tly remain unchanged.
F or the augmen tation v ariables at the non flip times ( ρ, ω ) N F the v ariational p osterior
densit y in Eq (9) for ∆ → 0 b ecomes
q (Π 1: N | z 0: T ) ∝
N
Y
i =1 Y
( t,ω ) ∈ Π i
Λ t,i ( z , ω ) exp  − Z T × R +
Λ t,i ( z , ω ) dω dt  , (17)
Π i = { ( t, ω ) } is a random p oin t set on the space T × R + where ρ t,i = 1. Eq (17) is
prop ortional to a densit y of a Poisson pr o c ess with resp ect to another homogeneous Poisson
pro cess on the same space (Konstan top oulos et al., 2011). Actually Eq (17) is an instance of
a marke d Poisson pr o c ess (Kingman, 1993), where ω are the ‘marks’ at times with ρ t,i = 1.
Λ t,i ( z t , ω ) = p PG ( ω | 1 , c t,i )Λ t,i ( z t ) is the in tensit y of this P oisson pro cess (see app endix A
and Donner and Opp er (2018)). The exp ectation ov er the P oisson pro cess is (Kingman,
1993)
E Q 
 X
( t,ω ) ∈ Π i
h ( t, ω )      
z 0: T 
 = Z T × R +
h ( t, ω )Λ t,i ( z , ω ) dt, (18)
where the exp ectation is conditioned on the tra jectory z 0: T . Note, that this result is similar
to the one obtained b y Donner and Opp er (2017). The difference is for Donner and Opp er
(2017) the P oisson pro cess rate w as piecewise constan t and hence the exp ectation can b e
written as sum o v er P oisson densities. Due to the exp ectation o v er the tra jectory z 0: T in
Eq. (17) this is not the case in the curren t work.
F or the recursiv e forward–bac kw ard Eqs (12) and (13) w e show that they become or-
dinary differen tial equations in the contin uous time limit ∆ → 0. The forward equations
are
∂ t f t ( z ) = X
z 0 6 = z
( ϕ ( z | z 0 ) f t ( z 0 ) − ϕ ( z 0 | z ) f t ( z )) − U t ( z ) f t ( z ) ,
and the bac kw ard equations
∂ t r t ( z ) = X
z 0 6 = z
ϕ ( z 0 | z )( r t ( z ) − r t ( z 0 )) + U t ( z ) r t ( z ) .
The limit of the U -function is
U t ( z ) = lim
∆ → 0 U t ( z , z 0 ) = −
N
X
i =1 
 X
( t 0 ,i 0 ) ∈F
δ ( t − t 0 ) δ i,i 0 ln ˆ p flip
t,i ( z )+Λ t,i ( z ) 

+  ϕ ( z | z ) + E Q \ 1 [ γ z ] (1 − E Q \ 1 [ π z ])  ,
with δ ( x ) b eing the Dirac delta function. F or the infinitesimal transition probabilities w e
deriv e
Q t ( z | z 0 ) ≈ δ z ,z 0 + dtg t ( z | z 0 ) ,
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with the transition rates
g t ( z | z 0 ) = ϕ ( z | z 0 ) r t ( z )
r t ( z 0 ) .
This results are equiv alen t to the one, which w ere obtained b y Opp er and Sanguinetti
(2008). F or detailed deriv ation of the forw ard–bac kw ard equations see app endix B. While
the differen tial equations ab o v e are exact in the limit, for practical in tegration sc hemes the
discrete up date (Eqs (12) and (13)) rules should b e preferred, b ecause they consider also
the non–linear terms in ∆, whic h guarantee f t and r t b eing non–negativ e.
Second factor The limit for the updates of the Gaussian p osterior ov er J 1: K is straigh t-
forw ard, since the sums b ecome in tegrals. F or the co v ariance matrix in Eq (14) b ecomes
Σ k ,i = 
 4 Z T X
t 0 ∈F i
δ ( t − t 0 ) C t E Q \ 2 [ ω t,i ] q t ( k ) dt
+4 Z T × R +
C t q t ( k )Λ t,i ( k , ω ) dω dt +  ˜
Σ k ,i  − 1  − 1
,
and equiv alen tly the me an in Eq (15) is
µ k ,i = Σ k
i 
 − Z T X
t 0 ∈F i
δ ( t − t 0 ) C t,i q t ( k ) dt + Z T
C t,i q t ( k )Λ t,i ( k ) dt +  ˜
Σ k ,i  − 1 µ J 
 .
F or the up date of state probabilities π 1: K nothing changes, except that gradien ts of the
lo w er b ound in Eq (7) in v olve expectations ov er P oisson pro cess defined b y Eq (18) instead
of Bernoulli v ariables.
Third factor The posterior o v er the state switc hing rate γ z remains a gamma densit y
in the limit, where the parameters of Eq (16) b ecome κ = R T P z g t ( z | z ) Q t ( z ) dt + 1 and
ν = R T P z Q t ( z ) dt (1 − E Q [ π z ]) + 1.
5. Results
T o ev aluate whether the deriv ed v ariational inference algorithm a go o d appro ximate p oste-
rior w e first generate data for N = 40 neurons with the generativ e mo del and compare the
inference results with the ground truth. Finally , w e fit the mo del to spiking data recorded
from monk ey V4 area in–vivo while the sub ject p erformed a p erceptual task.
Artificial Data W e first generate artificial data for with N = 40 neurons (see Fig 2).
The state probabilit y distribution in Fig 2 B is generated via a stick-br e aking pr o c ess with
α = 1. Note, that this generativ e pro cess do es not assume a finite n um b er of states K . The
external fields θ k are dra wn from a Gaussian N ( − 1 . 2 , 0 . 25), the couplings from a Laplace
distribution ( µ J = 0 , σ J = 0 . 5 / √ N ). The empirical probability of the resulting data for a
neuron b eing activ e is ∼ 0 . 15 whic h w ould corresp ond to an a v erage firing rate of ∼ 15 Hz.
The switc hing rate is set to γ z = 3 Hz and ∼ 16 . 5 min of data are generated. Mainly 4 states
are presen t the data, whose parameters J k are sho wn in Fig 2 A and state probabilities
are sho wn in Fig 2 B . 5s of example data are shown Fig 2 C . An activ e state of a neuron
is denoted b y blac k. The colouring sho ws at what times which state generated the data.
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89

Inference fr om non-st a tionar y spiking d a t a

[ms]

Figure 3: V ariational inference of the mo del from artificial data. The data are the
same as sho wn in Fig 2. A The lo w er b ound as th e n um b er of iterations of the
algorithm (outer lo op iterations). B The inferred state probabilities. C The true
couplings plotted against the inf e rr e d ones for eac h of the 4 states. The triangles
indicate the exte r nal fields, and circles inditcate the couplings. D The inferred
state lab e l distribution q ( z t ) (solid lines, colour indicates state iden tit y) with the
spiking data and true lab els again (coloured area) b elo w.
Practical implemen tation Before d is cussing the inference results, w e describ e the prac-
tical realisation of the inference algorithm. F or the i nitialisation w e assume that all states
are equal p robable at all times, i .e . Q ( z ( t ) = k ) = 1 /K for t ∈ T . The mean for the
couplings and external fields is set to the analytic solution for the assumption, that the
net w ork is uncoupled and the data are stationary . The standard deviation for the external
fields is set to σ θ = 1, resulting in a relativ ely broad prior. T o find the optimal v alue of
the scaling parameter, the mo del is fit once with σ J ∈ { 0 . 01 / √ N , 0 . 1 / √ N , 1 / √ N } and the
result with the maximal lo w er b ound is c hosen. α is set to 0 . 5.
After init ialisation, w e d e fi ne an inner and an outer lo op. In the inner lo op, all coupling
relev an t parameters (i.e. coupling J 1: K , augmen tation v ariables at flip times ω F , and the
laten t P oisson p ro cess Π 1: N ) are up dated un ti l the mean of couplings con v erges. Then in an
outer lo op the p osterior o v er the of state lab els z 0: t is up dated q ( z 0: T ), follo w ed b y the state
probabilities q ( π 1: K ), and the switc hi ng rate q ( γ z ). Then the inner lo op is rep eated. When
lo w er b ound con v erges, the pro cedure stops. Ho w ev er, to c hec k for con v e r ge n c e w e compare
the curren t v alue of lo w er b ound and the one from 10 iterations b efore. This is n e cessary ,
b ecause the algorithm often encoun ters a plateau, b efore finding a go o d configuration of
state lab els z 0: T (see Fig 3 A ).
Inference on artificial data F or the fit to artificial data, w e consider an in te grat ion
step of ∆ = 1 ms. The inference results are sho wn in Fig 3. The lo w er b ound is increasing
for eac h iteration (see Fig 3 A ) and the mean of the i nferred v ariational p osterior den-
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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sit y for the state probabilities π 1: K corresp onds to the underlying v alues (compare Fig 2 B
with 3 B ). F urthermore, the couplings of all 4 states are recov ered w ell (Fig 3 C ) and the
state lab els z 0: T in the data are inferred accurately (Fig 3 D ). The mean of the switc hing
rate E Q [ γ z ]=3 . 095 Hz is close to the ground truth. The fit with σ J = 0 . 1 / √ N yields the
maximal lo w er b ound.
V4 Monk ey Data After v alidating the inference algorithm w e analyse spiking data
recorded from monk ey V4. The data comprise activit y from 40 m ulti– and single units
recorded o v er 500 trials a 3 s. During eac h trial a either 0 ◦ or 90 ◦ drifting grating w as
presen ted to the monk ey for 2 s. The exp erimen t w as p erformed at the Universit y of
Pittsburgh. All exp erimen tal pro cedures w ere appro v ed b y the Univ ersit y of Pittsburgh
Institutional Animal Care and Use Committee, and w ere p erformed in accordance with the
United States’ National Institutes of Health (NIH) Guide for the Car e and Use of L ab or a-
tory A nimals . (F or details see Sn yder et al. (2015)). Exemplary data are sho wn in Fig 4 A .
F or the mo del inference all 500 trials are concatenated, i.e. the lac k of data b et w een trials
is ignored.
The inference results are sho wn in Fig 4. In Fig 4 A top and cen tre panel, we see that
state switc hes app ear shortly after stim ulus on and offset (trial a v eraged probabilit y of state
switc hing E trials [ P z (1 − ∆ g t ( z | z )) Q t ( z )] p eaks at 56 ms, not sho wn here). Stim ulus sp ecific
states can b e iden tified b y in v estigation of the a v eraged state lab el probabilities q ( z t ) ov er
90 ◦ and 0 ◦ trials, resp ectively (Fig 4 A Bottom). Ho wev er, there are differences b et w een
trials ev en under same stim ulus condition. Some trials sho w few state lab el switches during
the stim ulus phase (e.g. Fig 4 A T op). In con trast, other trials yield sev eral switches (e.g.
Fig 4 A Cen tre). Note, that these differences across trials can only b e seen b ecause the
mo del allo ws for cross–trial v ariability .
Since the states are defined b y the couplings and external fields we sho w the p osterior
mean of parameters for 4 differen t exemplary states (Fig 4 B ). The 1 st (blue) state from
the left is mainly inferred for p erio ds b efore and after the stim ulus (Fig 4 A ). Ho w ev er,
brief irregularly app earing p erio ds during stim ulus presen tation with collectiv e sparse firing
are also assigned to this state. The 2 nd (dark green) state frequen tly app ears shortly after
stim ulus onset. The last to states are stim ulus selectiv e (3 rd (la vender) for 90 ◦ and 4 th
(orange) for 0 ◦ ). The inferred mean of the inferred p osterior o v er the couplings J k ,ij sho ws
sparse connectivit y structures among all states (i.e. most J k ,ij ≈ 0). Ho w ev er, for the 1 st
(blue) state w e observe more positive couplings than the last t w o (la v ender and orange)
stim ulus states indicating more uncorrelated activit y with the stim ulus presen t.
T o sho w why considering single–trial analysis is imp ortan t we focus on the stimulus
p erio d (from 200 ms to 2 s, to exclude the transien t at stim ulus onset). W e align the
p opulation a v eraged activit y data during this p erio d to the onsets inferred for eac h of the
4 states sho wn in Fig 4 B . A state onset is defined as the time p oin t a state b ecomes the
most lik ely one. W e a v erage o v er all inferred state onsets, resulting in the state–sp ecific
a v erage p opulation activit y (Fig 4 C T op). F or the 1 st state we observ e a strong drop in the
fraction of activ e units after stimulu s onsets. The 2 nd state sho ws a drop b efore the state
onset (lik ely caused b y the frequen tly preceding 1 st state) follo w ed b y a strong increase in
p opulation activit y . The stim ulus states sho w a brief activit y increase. Additionally , the
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Inference fr om non-st a tionar y spiking d a t a

Neuron ID
9 0
◦
0
◦
Neuron ID
0 2000
Time [ms]
0
1
q
(
z
(
t
))
0 2000
Time [ms]
0
1
Neuron ID
θ i
Neuron ID
J ij θ i
Neuron ID
J ij θ i
Neuron ID
J ij θ i
Neuron ID
J ij
3 2 1 0
θ i
0.5 0.0 0.5
J ij
0.05
0.10
0.15
Frac. active units
200 0 200
100
0
100
LFP
200 0 200 200 0 200 200 0 200
Time from state onset [ms]
A B
C

Figure 4: Mo del fit to V4 monk ey data 500 trials recorded in–viv o while the monk ey
p erformed a p erceptual task. Spiking data of 40 units are analysed. A Left: Tw o
individual tr ials where the stim ulus is a 90 ◦ drifting grating. Colours indicate
the iden tit y of the most lik ely state z t at eac h time p oin t. The lo w er ro w sho ws
the state lab el probabilities q ( z t ) a v e raged o v er all 90 ◦ trials. Righ t: Same as l e ft
for trails where stim ulus is a 0 ◦ drifting gratin g. B P osterior mean of extern al
fields and couplings for 4 differen t states. F rom left to r igh t: The first (b lue)
state is p rominen t during non-stim ulus p e r io ds and du ring stim ulus p erio ds when
the net w ork exhibits collectiv e ly sparse activit y . The second (dark green) state
app ears most frequen tly after stim ulu s onset. T h e last t w o states are stim ulus
selectiv e (3 rd (la v ender) for 90 ◦ and 4 th (orange) for 0 ◦ ). C Av erage p opulation
activit y (top) and the LFP (b ottom) aligned with the state onsets. Only p erio ds
that app eared b et w een 0 . 2 s and 2 s after stim ulus on s ets are considered.
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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a v eraged lo cal field p oten tial (LFP) - recorded sim ultaneously with the spik e data - sho ws
a strong signature of the 1 st state as well as for the 2 nd state (Fig 4 C Bottom).
6. Discussion
T o statistically describ e non–stationary data recorded from m ultiple neurons w e presen ted
a mo del, whic h considers couplings among recorded neurons and do es not require discreti-
sation of the observ ed data. The prop osed v ariational algorithm infers hidden states in an
unsup ervised manner, and hence forms a flexible analysis framew ork for these data.
W e utilised the augmen tation scheme of (Donner and Opper, 2017) to render the data
lik eliho o d of the kinetic Ising mo del in to a Gaussian form with resp ect to the mo del param-
eters J 1: K . F urthermore, the Laplace prior is written as a Gaussian mixture mo del. This
allo w ed for the deriv ation of an efficien t v ariational mean–field algorithm, where, apart from
the state–probabilities, all up dates can b e computed analytically .
F urthermore, the augmen ted form of the mo del allo ws for alternativ e inference schemes.
F or example, an empirical Bay es approac h is p ossible, where one deriv es an exp ectation–
maximisation (EM) algorithm (Dempster et al., 1977). F or this w e consider only the p oin t
estimate of the couplings J 1: K , the state probabilities π 1: K , and the state jump rate γ z .
This allo ws us to define the w ell–kno wn Q-function for the mo del, whic h can iterativ ely b e
maximised. In the E–step one computes analytically the p osterior o v er state–lab els z 0: T
and augmen ted v ariables ω F , ( ρ, ω ) N F , β 1: K . In the M–step the Q–function is optimised
with resp ect to the v ariables J 1: K , π 1: K , γ z . F or the couplings J 1: K one has only to solv e
linear system of equations. Since π 1: K , γ z are coupled to eac h other, one requires some
n umerical optimisation metho d to maximise the Q-function, e.g. gradien t ascent or second
order metho ds. This algorithm is then guaran teed to con v erge to a (lo cal) maxim um of the
non–augmen ted (p enalised) mo del lik eliho o d.
Alternativ ely the augmen tations allo w for a Mark o v c hain Mon te Carlo sc heme. The
v ariables J 1: K , ω F , ( ρ, ω ) N F , γ z can b e sampled directly from their conditional p osterior.
F or the sparsit y v ariables β 1: K the conditional p osterior is – as in the v ariational case – a
generalised in v erse Gaussian, whic h can b e sampled via efficien t rejection sampling (A tkin-
son, 1982). Another option is to assume a spike–and–slab prior, which can be sampled
efficien tly (Linderman et al., 2016). F or the state probabilities π 1: K one could in principle
assume a Chinese restauran t pro cess prior, such that the n um b er of states K do es not need
to b e fixed. Then the state lab els z 0: T and probabilities π 1: K could b e sampled b y a MCMC
pro cedure (Stim b erg et al., 2012).
The augmen tation sc heme utilised in this w ork can b e applied to generalised linear
mo dels (GLMs) with p oin t pro cess lik eliho o d ha ving an in tensit y function of the form
Λ t = γ σ ( h t ( · )), where σ ( · ) is the sigmoid link function and h t ( · ) dep ends on observed co v ari-
ates and linearly on the mo del parameters. F or mo dels with such likelihoo ds and Gaussian
priors, tractable p osteriors o ver the model parameters can b e obtained efficiently (Donner
and Opp er, 2018). The GLM with sigmoid link function has b een sho wn to b e adv an tageous
for describing spiking activit y compared to the more classical choice Λ t = exp( h t ( · )) (Capone
et al., 2018). The consideration of contin uous time allo ws to a v oid the question of c ho osing
the bin–size for time discretisation. Consequen tly , optimisation of this parameter is not
required, as in contrast to more traditional paradigms. Admittedly , while preserving the
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Inference fr om non-st a tionar y spiking d a t a
temp oral precision of the data, the difficult c hoice of bin size is shifted to the question of
ho w long a neuron is considered to b e ‘activ e’ after a spike. This can b e circum v en ted with
GLMs that ha v e parametrised history k ernels. Here w e fo cused on the kinetic Ising mo del
b ecause it allo ws in terp olating b et w een discrete and contin uous time (Zeng et al., 2013).
The metho dology presen ted here can also b e used for mo dels with con tin uous dynamics.
The augmen tation scheme can straigh tforw ardly b e applied to the Gaussian pr o c ess factor
analysis (GPF A) (Y u et al., 2009). T o preserv e tractabilit y of the mo del Y u et al. (2009)
assumed, that the observed square root spike–coun t is Gaussian distributed, while a Poisson
distribution w ould b e a more natural c hoice. While the treatmen t of P oisson lik eliho o ds for
discrete time (Nam, 2015) and P oin t pro cess lik eliho o ds for con tin uous time (Dunc k er and
Sahani, 2018) w ere already addressed, the augmentations dev elop ed in Donner and Opp er
(2017) and the presen t w ork probably allo w for efficien t inference sc hemes for the GPF A
mo del.
Non–stationary mo dels as the one presen ted here allo w for statistical description of
single–trial data. In exp erimen ts with rep etitiv e trial structure we ha v e demonstrated that
our mo del can unco v er v ariations across trials. Not taking these in to accoun t migh t lead to
spurious results Bro dy (1999).
In conclusion, this w ork represen ts a step to w ards in v estigating recorded neural net w orks
in con tin uous time when kno wledge of the underlying dynamics is scarce.
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
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App endix A. Con tin uous time limit for p osterior of laten t v ariables at
non flip times
W e w ant to deriv e the limit ∆ → 0 of the v ariational p osterior densit y
q (( ρ, ω ) N F | z 0: T ) =
N
Y
i =1 Y
t ∈N F i
[ p PG ( ω t,i | 1 , c t,i ( z ))] ρ t,i Ber( ρ t,i | P ( ρ )
t,i ( z )) .
First, w e note that in the limit ∆ → 0 becomes the whole space N F i → T . W e define a
set Π i = { ( t, ω ) } con taining all the time p oin ts where ρ t,i = 1 and write
q (( ρ, ω ) N F | z 0: T ) =
N
Y
i =1 Y
( t,ω ) ∈ Π i
p PG ( ω t,i | 1 , c t,i ( z )) P ( ρ )
t,i ( z ) Y
( t,ω ) 6∈ Π i
(1 − P ( ρ )
t,i ( z )) .
The con tin uous time limit of this expression is
q (Π 1: N | z 0: T ) ∝
N
Y
i =1
Λ t,i ( z , ω ) exp  − Z T × R +
Λ t,i ( z , ω ) dω dt  ,
whic h is prop ortional to a pro duct of P oisson pro cess densities o v er the space T × R + , where
the densities are defined with resp ect to a homogeneous P oisson pro cess measure (Konstan-
top oulos et al., 2011).
App endix B. Deriv ation of forw ard–bac kw ard equations for a Mark o v
jump pro cess
In this section w e sho w ho w to deriv e the forw ard–bac kward equations for a Mark o v jump
pro cess. As w e will show, the final results are the same as in Opp er and Sanguinetti (2008).
While Opp er and Sanguinetti (2008) consider the limit ∆ → 0 from the start w e deriv e the
forw ard–bac kw ard equations first for a Mark o v c hain and then tak e the limit in the end.
In practice, a finite ∆ needs to b e c hosen and the subsequen t deriv ations yield a preferable
in tegration sc heme. As sho wn the v ariational p osterior densit y has the form
Q ( z 0: T ) ∝ ˜
Q ( z 0: T ) = Y
t ∈T
P 1 ( z t | z t +∆ ) exp ( − U ( z t , z t − ∆ )∆) P ( z 0 ) ,
W e w ant to determine the Mark o v c hain factors
Q ( z 0: T ) = Y
t ∈T
Q t ( z t | z t − ∆ ) Q 0 ( z 0 ) .
The ob jectiv e function w e minimise is the Kullbac k–Leibler div ergence b et w een Q and the
unnormalised ˜
Q as
D KL  Q k ˜
Q  = E Q  ln Q
˜
Q  = X
t ∈T X
z ,z 0
Q t ( z | z 0 ) Q t − ∆ ( z ) ln Q t ( z | z 0 )
P 1 ( z | z 0 ) exp  − ˜
U t ( z , z 0 )∆ 
+ X
t ∈T X
z
Q t ( z ) U data
t ( z )∆ + X
z
Q 0 ( z ) ln P 0 ( z )
Q 0 ( z ) .
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95

Inference fr om non-st a tionar y spiking d a t a
The factors of the Mark ov c hain need to fulfil the marginalisation constrain ts
1 = X
z 0
Q t ( z 0 | z ) (19)
Q t ( z ) = X
z 0
Q t ( z | z 0 ) Q t − ∆ ( z 0 ) . (20)
F or the momen t we only consider only the constrain t in Eq (20) and will tak e care of the
one in Eq (19) during the deriv ations. The constrain t optimisation problem is
L z = D KL  Q k ˜
Q  + X
t ∈T X
z
λ t ( z ) Q t ( z ) − X
z 0
Q t ( z | z 0 ) Q t − ∆ ( z 0 ) ! , (21)
where λ t ( z ) are the Lagrangian m ultipliers. T aking the deriv ative with respect to Q t ( z | z 0 ),
setting it to 0, and considering the normalisation constrain t Eq (19) we deriv e
Q t ( z | z 0 ) =
P 1 ( z | z 0 ) exp  − ˜
U t ( z , z 0 )∆ + λ t ( z ) 
P z P 1 ( z | z 0 ) exp  − ˜
U t ( z , z 0 )∆ + λ t ( z )  . (22)
W e tak e the deriv ativ e with resp ect to Q t ( z t ) of Eq (21) and equate it to 0
∂ L z
∂ Q t ( z ) = X
z 0
Q t +∆ ( z 0 | z )  − ln P 1 ( z | z 0 ) + ˜
U t +∆ ( z 0 , z )∆ + ln Q t +∆ ( z 0 | z ) − λ t +∆ ( z 0 ) 
+ λ t ( z ) + U data
t ( z t )∆ = 0 .
By inserting (22) in to this result and defining r t ( z ) = e λ t ( z ) w e get the backw ard equations
r t ( z ) = X
z 0
P 1 ( z 0 | z , ϕ 1: K , ∆) exp  − U t +∆ ( z 0 , z )∆  r t +∆ ( z 0 ) . (23)
This allo ws us already to solv e of the inferring the distribution o v er the Mark o v Chain since
w e can first recursiv ely calculate r t . W e then solve for the marginals b y iterating forw ard
Q t ( z ) = X
z 0
Q t ( z | z 0 ) Q t − ∆ ( z 0 ) . (24)
Since w e consider quite long Marko v c hains, we w ould lik e to decouple these forw ard–
bac kw ard iterations, suc h that they can b e parallelised. In order to do so, we define
Q t ( z ) ∝ f t ( z ) × r t ( z ). Substituting this in to Eq (24), and with Eqs (23) and (22) w e
deriv e the forw ard equations
f t ( z ) = X
z 0
P 1 ( z | z 0 , ϕ 1: K , ∆) exp  − U t ( z , z 0 )∆  f t − ∆ ( z 0 ) , (25)
whic h do not dep end on r t .
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Chapter 8. Bayesian network infer enc e fr om non-stationary spiking data
96

Donner and Opper
Con tin uous time limit F or ∆ → 0 the bac kw ard and forw ard equations (23) and (25)
b ecome differen tial equations of the form
∂ t r t ( z ) = X
z 0 6 = z
ϕ ( z 0 | z )( r t ( z ) − r t ( z 0 )) + U t ( z ) r t ( z ) ,
∂ t f t ( z ) = X
z 0 6 = z
( ϕ ( z | z 0 ) f t ( z 0 ) − ϕ ( z 0 | z ) f t ( z )) − U t ( z ) f t ( z ) ,
where U t ( z ) = lim ∆ → 0 U t ( z , z 0 ). F or these results w e neglected the higher order term
O (∆ 2 ) , O (∆ 3 ) , etc. The limit for the transition probabilit y in Eq (22) b ecomes
Q t ( z | z 0 ) ≈ δ z ,z 0 + dtg t ( z | z 0 ) , (26)
where
g t ( z | z 0 ) = φ ( z | z 0 ) r ( z )
r ( z 0 ) .
In terestingly , the master e quation of a Marko v jump pro cess can b e deriv ed from Eq (26)
and the limiting case of the constrains in Eqs (19) and (20) b eing
∂ t Q t ( z t ) = X
z 0 6 = z
g t ( z | z 0 ) Q t ( z 0 ) − g t ( z 0 | z ) Q t ( z ) .
This constrain t is considered from the start b y Opp er and Sanguinetti (2008). While from a
theoretical p oin t of view the limiting equations come out v ery elegan t, for practical reasons,
an in tegration sc heme using Eqs (23) and (25) is preferred, b ecause these equations consider
also higher order terms of the differen tial equations of the finite in tegration step size ∆ and
ensure, that r t and f t are alw a ys non–negativ e.
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Chapter 9
Unpublished article: Inferring the
c ol le ctive dynamics of neur onal
p opulations fr om single-trial spike
tr ains using me chanistic mo dels
Authors:
Christian Donner 1 , 2 , Manfred Opp er 1 , 2 , Josef Ladenbauer 2 , 3
1
T ec hnisc he Universität Berlin.
2
Bernstein Cen ter for Computational Neuroscience Berlin.
3
École Normale
Sup érieure, P aris.
Chapter 9
This c hapter comprises the unpublished manuscript, whic h is authored b y m yself
(CD), Prof. Manfred Opp er (MO), and Dr. Josef Ladenbauer (JL).
Con tributions :
CD and JL conceiv ed and designed the w ork with help of MO. CD deriv ed the inference algorithms
and dev elop ed the Python co de. CD p erformed the n umerical exp erimen ts. CD wrote the manuscript
with substan tial con tribution from JL.
Python co de on GitHub: https://gith ub.com/c hristiando/doubly_sto c hastic_lif_inference.git
103

Inference with neur onal mechanistic models
Inferring the collectiv e dynamics of neuronal p opulations
from single-trial spik e trains using mec hanistic mo dels
Christian Donner [email protected]
Manfred Opp er [email protected]
A rtificial Intel ligenc e Gr oup
T e chnische Universit¨ at Berlin
Berlin, Germany
Josef Laden bauer josef.ladenba [email protected]
L ab or atoir e de Neur oscienc es Co gnitives et Computationnel les
´
Ec ole Normale Sup´ erieur e - PSL R ese ar ch University
Paris, F r anc e
Abstract
Multi-neuronal spik e-train data recorded in–vivo often exhibit ric h dynamics as w ell as con-
siderable v ariability across cells and repetitions of identical experimental conditions (trials).
Efforts to c haracterise and predict the p opulation dynamics and the con tributions of indi-
vidual neurons require mo del-based to ols. Abstract statistical mo dels allo w for principled
parameter estimation and mo del selection, but p ossess only limited in terpretive pow er b e-
cause they t ypically do not incorp orate prior biophysical constrain ts. Here w e present a
statistically principled approac h based on a p opulation of doubly-sto c hastic integrate-and-
fire neurons, taking in to account basic bioph ysics. This mo del class comprises an idealised
description for the dynamics of the neuronal mem brane voltage in response to fast indepen-
den t and slow er shared input fluctuations. T o efficiently estimate the model parameters and
compare differen t mo del v arian ts w e compute the lik eliho o d of observ ed single-trail spike
trains b y leveraging analytical methods for spiking neuron mo dels com bined with infer-
ence tec hniques for hidden Marko v mo dels. This allows us to reconstruct the shared input
v ariations, classify their dynamics, obtain precise spike rate estimates, and quan tify how
individual neurons couple to the lo w-dimensional ov erall p opulation dynamics, all from a
single trial. Extensiv e ev aluations based on sim ulated data sho w that our metho d correctly
iden tifies the dynamics of the shared input pro cess and accurately estimates the mo del
parameters. V alidations on ground truth recordings of neurons in–vitro demonstrate that
our approac h successfully reconstructs the dynamics of hidden inputs and yields improv ed
fits compared to a t ypical phenomenological mo del. Finally , we apply the method to a
neuronal p opulation recorded in–viv o, for which w e assess the con tributions of individual
neurons to the o verall spiking dynamics. Altogether, our w ork pro vides statistical infer-
ence to ols for a class of reasonably constrained, mec hanistic mo dels and demonstrates the
b enefits of this approac h to analyse measured spike train data.
Keyw ords: non–stationary spiking activit y , leaky in tegrate and fire neuron, hidden
Mark ov model, Ornstein–Uhlenbeck process, Marko v jump pro cess
1

Chapter 9. Inferring the c ol le ctive dynamics of neur onal p opulations fr om
single-trial spike tr ains using me chanistic mo dels
104

Donner et al.
1. In tro duction
Cortical computations are represen ted in the collectiv e spiking activit y of m ultiple neu-
rons. The gro wing in terest to unco v er how these neural populations pro cess and transform
(complex) incoming information in to decisions and motor actions has brough t ab out cell-
resolving activit y measurements at an increasing pace and scale. Muc h information app ears
to b e con tained in p opulation dynamics of lo w dimensionality (Shadlen and Newsome, 1998;
Luczak et al., 2009; Man te et al., 2013; Aljadeff et al., 2016). T o in terpret the measured
spik e trains statistical metho ds based on generativ e mo dels can b e v ery p o werful, pro vid-
ing also the opp ortunit y to mak e predictions for unobserv ed conditions. Often parametric
phenomenological mo dels are fitted to these data and used for analysis. T ypical examples
include mo dels assuming P oisson data (Brillinger, 1988; Latimer et al., 2015; Mac k e et al.,
2015), Ising and generalised linear mo dels (Gaudino et al., 2014; Shimazaki et al., 2012;
Chic hilnisky, 2001; T ruccolo, 2004; Pillow et al., 2008; Aljadeff et al., 2016). F urthermore,
explicit dimension reduction metho ds that extract a lo w dimensional tra jectory of observ ed
spiking data, w ere prop osed (Cunningham and Y u, 2014; P andarinath et al., 2018). Mo dels
of that kind are a flexible mo del class, can b e efficien tly fit, and are w ell suited to capture
statistical prop erties of the data. A dra wback of these mo dels is, how ev er, that their pa-
rameters do not directly relate to the underlying bioph ysics, which limits their in terpretiv e
p o w er.
Mec hanistically more detailed mo dels describ e the dynamics of the neuronal mem brane
v oltage, whic h is t ypically not observ ed. The most classical, and probably b est kno wn
mo del is of the Ho dgkin–Huxley t yp e (Ho dgkin and Huxley, 1952), which con tains n umerous
parameters and a set of partial differen tial equations, that mak e fitting already c hallenging
kno wing the mem brane v oltage trace (Luec kmann et al., 2017; Meliza et al., 2014). An
alternativ e, prominen t class of idealised mo dels are of the inte gr ate-and-fir e (I&F) type.
These mo dels ha v e b ecome state-of-the-art for describing neural activit y in in–viv o lik e
conditions (i.e. not kno wing the in ternal v oltage dynamics)(Joliv et et al., 2008; Badel et al.,
2008; Gerstner and Naud, 2009; Harrison et al., 2015; P ozzorini et al., 2015; T eeter et al.,
2018) and ha v e b een applied in a m ultitude of studies on neural netw ork dynamics. While
bioph ysically more faithful than purely phenomenological mo dels, these spiking neuron
mo dels are also more complex to fit, particularly in the presence of unknown, noisy and
correlated inputs, ha ving only access to the spike times as t ypical for in–viv o data.
T o accoun t for the collectiv e sto c hastic spiking dynamics observ ed in–vivo w e consider a
p opulation of doubly-sto c hastic I&F neurons. Their hidden inputs con tain fast indep enden t
fluctuations that giv e rise to spiking v ariability across neurons and trials, and slo w er shared
v ariations due to common driv e, whic h dominate the lo w-dimensional ov erall p opulation
dynamics. W e dev elop a statistically principled approac h to fit this t yp e of mo del to single-
trial spik e trains and allo w for a quan titativ e comparison b et w een differen t mo del v ariants,
including simpler phenomenological mo dels, according to established criteria. Specifically ,
w e efficien tly compute the lik eliho o d of a giv en spike train b y exploiting analytical metho ds
for spiking neuron mo dels (Laden bauer et al., 2018) combined with inference tec hniques
for hidden Mark o v mo dels (Rabiner, 1989). Based on sim ulated data w e then ev aluate this
approac h extensiv ely in terms of reconstruction of the true dynamics, classification of their
t yp e (con tin uous and jump y input dynamics), and estimation of the mo del parameters.
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