Zenodo doi:10.5281/zenodo.17430468
Pe sonalized p ognosis & ea men
wi h Bayesian nonpa ame ic in e ence
An example s udy on con e sion
om Mild Cogni i e Impai men o Alzheime ’s Disease
P.G.L. Po a Mana
Wes e n No way Uni e si y o Applied Sciences <pgl po amana.o g>
I. Rye
Uni e si y o Oslo
A. Vik
Haukeland Uni e si y Hospi al,
Be gen
M. Kociński
Uni e si y o Be gen
A. Lunde old
Mohn Medical Imaging and Visualiza ion Cen e (MMIV), Depa men o Radiology, Haukeland
Uni e si y Hospi al, Be gen
Uni e si y o Be gen
A. J. Lunde old
Depa men o Biological and Medical Psychology, Uni e si y o Be gen
A. S. Lunde old
Mohn Medical Imaging and Visualiza ion Cen e (MMIV), Depa men o Radiology, Haukeland
Uni e si y Hospi al, Be gen
Wes e n No way Uni e si y o Applied Sciences
18 No embe 2022; upda ed 23 No embe 2025 [d a ]
The p esen wo k p esen s a s a is ically sound, igo ous, and model- ee in-
e ence me hod o use in pe sonalized medicine, oge he wi h a so wa e
implemen a ion. The me hod, Bayesian nonpa ame ic in e ence, is designed
i s o lea n om a se o clinical da a wi h ele an p edic o s and p edic ands,
and hen o assis a clinician in he assessmen o p ognosis & ea men o
new pa ien s. I allows he clinician o inpu , o each new pa ien , addi ional
pa ien -dependen clinical in o ma ion, as well as pa ien -dependen in o ma ion
abou bene i s and d awbacks o a ailable ea men s. We apply his me hod and
so wa e in a ealis ic se ing o clinical decision-making, inco po a ing clinical,
en i onmen al, imaging, and gene ic da a, using a da a se o subjec s su e ing
om mild cogni i e impai men and Alzheime ’s Disease. We show how he
me hod is heo e ically op imal, and discuss some o i s majo ad an ages o
decision-making unde isk, esou ce planning, impu a ion o missing alues,
assessing he p ognos ic impo ance o p edic o s, and u he uses.
Keywo ds: Clinical decision making, U ili y heo y, P obabili y heo y, Bayesian nonpa ame ics,
Machine Lea ning, A i icial In elligence, Base- a e allacy
1
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
1 In oduc ion: Pe sonalized p ognosis, ea men , s a is ics,
and assis i e so wa e
1.0 P ologue: Fou unique pa ien s
Mee Oli ia, A iel, Bianca, Cu is.
1
These ou pe sons don’ know each
o he , bu hey ha e some hing in common: hey all su e om a mild
o m o cogni i e impai men , and a e a aid ha hei impai men will
u n in o Alzheime ’s Disease wi hin a couple o yea s. This is why
each o hem ecen ly unde wen a wide ange o clinical examina ions
and es s, including b ain imaging. Today hey a e ecei ing he esul s.
Based on hei indi idual esul s, on a ailable clinical s a is ical da a,
and on o he ele an in o ma ion, hei clinician will assess hei isk o
de eloping Alzheime ’s Disease. Then, oge he wi h he pa ien s and
hei ela i es, he clinician will make a decision among ou dis inc
p e en i e- ea men op ions, a ailable o each pa ien .
2
In hese asks,
he clinician will be helped by an assis i e in e ence me hod and so wa e.
Besides a sha ed diagnosis o Mild Cogni i e Impai men and asso-
cia ed wo ies, hese pa ien s ha e o he hings in common – bu also
some di e ences. Le ’s ake Oli ia as e e ence, and lis he simila i ies
and di e ences be ween he and he o he h ee pa ien s:
•
Oli ia and A iel ha e iden ical esul s on he clinical and labo a o y
measu es and age. They would also incu simila bene i s and losses
om he ou a ailable ea men op ions. A iel, howe e , comes
om a di e en geog aphical egion, which p esen s a highe a e o
con e sion om Mild Cogni i e Impai men o Alzheime ’s Disease.
And unlike Oli ia, A iel comes om a amily wi h a hea y his o y
o Alzheime ’s Disease. Because o his geog aphical and amily
backg ound and some ele an s a is ics ound in some publica ions,
he clinician judges, be o e seeing he clinical da a, ha he e’s a
65% p obabili y ha A iel’s cogni i e impai men will con e o
Alzheime ’s Disease.
•
Oli ia and Bianca ha e iden ical clinical esul s and age; hey also
come om he same geog aphical egion and ha e e y simila
1
These a e pu ely ic i e cha ac e s bu wi h clinically ealis ic condi ions; any e e ence
o eal pe sons is pu ely coinciden al.
2
In he p esen pape we use “p ognosis” in
a gene al sense o include also “diagnosis”, and “ ea men ” qui e loosely o mean any
cou se o ac ion a clinician migh ake, including p e en i e ea men o e en “addi ional
es s”.
2
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
amily his o ies. In ac , we shall see ha hey ha e he same
p obabili y o de eloping Alzheime ’s Disease. Bianca, howe e ,
su e s om se e al alle gies and addi ional clinical condi ions ha
ende some o he ea men op ions sligh ly iskie o he .
•
Oli ia and Cu is ha e di e en esul s on all measu es included in
he clinical and labo a o y examina ions; Oli ia is also mo e han
10 yea s olde han Cu is. They o he wise come om he same
geog aphical egion, ha e e y simila amily his o ies, and would
incu simila bene i s o losses om he ea men op ions. No e
ha he imaging esul o Cu is (hippocampal olume) is missing.
Conside ing he simila i ies and di e ences among hese pa ien s,
which o he ou a ailable ea men s will be op imal o each o hem?
The clinician will ind ha , despi e he many ac o s in common among
ou ou pa ien s – e en despi e Oli ia’s, A iel’s, and Bianca’s iden ical
clinical esul s, and Oli ia’s and Bianca’s iden ical p obabili y o con-
e sion o Alzheime ’s Disease – he op imal ea men o each pa ien is
di e en om hose o he o he h ee – how come?
1.1 Assis i e in e ence so wa e: pe sonalized inpu and ou pu
In he example abo e, we said “in hese asks, he clinician will be
helped by an assis i e in e ence me hod and so wa e”. The need o such
compu a ional help is clea om he as amoun o clinical s a is ical da a
and he la ge numbe o clinical p edic o s oday a ailable o clinicians.
Bu how should such assis i e me hod and so wa e be designed in o de
o ake ully in o accoun pa ien di e ences?
Al hough he example abo e conce ns spe-
ci ically Alzheime ’s Disease, he di e ences
among pa ien s desc ibed he e apply mo e
gene ally o mos , i no all, clinical p oblems o
p ognosis and ea men . These di e ences can
be b oadly ca ego ized as “di e ence in aux-
ilia y o supplemen a y es s and backg ound
in o ma ion” (Oli ia s A iel), “di e ence in
bene i and a ailabili y o ea men s” (Oli ia s Bianca), “di e ence in
clinical p edic o s” (Oli ia s Cu is), as schema ized in he side igu e.
Each o hese di e ence ca ego ies can a ec he clinician’s inal choice o
3
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
op imal ea men . An assis i e me hod and so wa e should he e o e
e lec hese di e ences in i s inpu , i s ou pu , o bo h:
•
In p inciple, he e could be h ee kinds o inpu “slo s”, whe e he
clinician can inpu he cu en pa ien ’s speci ic alues as ega ds
clinical p edic o s, auxilia y in o ma ion, and ea men op ions &
bene i s.
•
I inpu slo s a e only a ailable o one o wo o he ca ego ies
abo e, he ou pu should a leas be o such a kind as o allow he
clinician o in eg a e he cu en pa ien ’s speci ic alues o he
missing inpu ca ego ies.
To app ecia e hese equi emen s, one should con as he inpu and
ou pu o many kinds o machine-lea ning classi ica ion algo i hms.
These ypically only allow he inpu o a pa ien ’s clinical p edic o s, wi h
no space o pa ien -speci ic auxilia y in o ma ion o o adjus men s
o di e ences in backg ound s a is ics ( hink o Oli ia s A iel). And
hey ypically ou pu only a disc e e p ognos ic label (say, “s able Mild
Cogni i e Impai men ” s “con e sion o Alzheime ’s Disease”), bu no
measu e o he unce ain y abou ha label. Un o una ely, such ou pu
does no allow he clinician o assess ea men bene i s and losses o he
cu en pa ien , o his assessmen depends no on he p esence (p esen
o u u e) o a disease, bu on he isk o i s p esence. We shall discuss
hese poin s a leng h in §§ 3.2 and 3.3.
The pu pose o he p esen wo k is o p esen an assis i e me hod,
Bayesian nonpa ame ic in e ence
3
, oge he wi h a so wa e implemen a-
ion, which mee he equi emen s abo e. This me hod is designed o
i s lea n om a da ase o clinical da a wi h ele an p edic o s and
p edic and
4
, and hen assis a clinician in he assessmen o p ognosis &
ea men o new pa ien s. I o e s hese en ea u es:
1.
I can wo k wi h clinical p edic o s comp ising any combina ion
o ca ego ical and one-dimensional (con inuous, disc e e o dinal,
unbounded o bounded, uncenso ed o censo ed) a ia es. The
3
Mo e p ecisely Bayesian nonpa ame ic densi y in e ence; see e.g. Rod íguez e al. 2009;
Bha acha ya & Dunson 2010; and Walke ’s 2010 wi y o e iew.
4
li e ally “quan i y o
be p edic ed” o , mo e gene ally, in e ed (c . measu and in me ology, jcgm 2012 2.3). We
ind his e m, used in me eo ology and clima e science, mo e p ecise and less obscu e o
misleading han “dependen a ia e”, “ esponse a ia e”, “ou come a iable”, o simila .
4
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
p edic and can also be any combina ion o ca ego ical and one-
dimensional a ia es.
2.
I ea s p edic o and p edic and a ia es on equal oo ing, in he
sense ha he clinician can a any momen decide o in e some
o he a ia e gi en he es .
3.
I does no equi e ha he cu en pa ien be conside ed in all
espec s as a membe o he popula ion unde lying he lea ning
da ase . The pa ien can be conside ed a membe only condi ionally
on pa icula a ia e alues.
4. I accep s h ee inpu s:
(i1) he clinical-p edic o alues o he cu en pa ien ;
(i2)
in o ma ion abou which p edic and-p edic o ela ionships
lea ned om he da ase can be gene alized o he cu en pa-
ien , and a p io p ognos ic p obabili y ep esen ing auxilia y
in o ma ion;
(i3)
a se o ea men op ions and hei bene i s and losses o he
cu en pa ien .
5. I yields h ee basic ou pu s:
(o1)
any p ognos ic p obabili ies o likelihoods abou p edic o s
and p edic and desi ed by he clinician, gi en inpu (i1);
(o2) inal p ognos ic p obabili ies, gi en inpu s (i1)–(i2);
(o3) op imal ea men , gi en inpu s (i1)–(i3);
6.
I s inpu and ou pu s a e modula , in he sense ha he clinician
can, o ins ance, gi e inpu s (i1)–(i2) only, ge a p ognos ic p obab-
ili y (o2) as ou pu , and hen p oceed o ea men assessmen by
o he means o me hods.
7.
I wo ks e en i p edic o da a a e missing, bo h in he lea ning
da ase and o he cu en pa ien .
8.
I can quan i y he unce ain y o i s own ou pu s, allowing o
sensi i i y analyses. Fo example, i can ell how much a p ognos ic
p obabili y could ha e been di e en i he lea ning da ase had
been la ge , o whe he he op imal ea men could be di e en i a
pa icula missing p edic o o he cu en pa ien we e a ailable.
5
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
9.
I can make a ious kinds o long- e m o ecas s, such as equency
o p ognoses wi h gi en p obabili ies, equency o p esc ibed
ea men s, and simila – p o ided ha he da ase used o i s
lea ning can be conside ed ep esen a i e o he ull popula ion.
10.
I is model- ee and ex ac s he maximal amoun o in o ma ion he-
o e ically con ained in he lea ning da ase , and he e o e achie es
he maximal p ognos ic powe ha he p edic o s can yield. In o he
wo ds, i is unbea able.
Le us commen on some o hese ea u es. We belie e ha he
capabili y o wo king wi h complex p edic ands, ea u e 1., is impo an
o a mo e ealis ic and nuanced app oach o p ognosis. In he case
o Alzheime ’s Disease, o ins ance, a simple dicho omy “has disease”
s. “doesn’ ha e disease” is possibly an o e simpli ica ion
5
. Wi hou
ea u e 3., he capabili y o auxilia y con ex ual in o ma ion, he me hod
would be o no use in he o en occu ing case o pa ien s ha ing peculia
clinical con ex s. The capabili y o dealing wi h missing da a, ea u e 7.,
is impo an o a conc e e implemen a ion in a clinical se ing, ypically
a lic ed by impu a ion p oblems. Fea u e 8. is ex emely impo an
o a clinician o assess he eliabili y o inal decisions and hones ly
in o m he pa ien o he possibili y o unwan ed ou comes. Finally,
ea u es 2. and 10., he ac ha his me hod yields he maximal amoun o
in o ma ion join ly con ained in all a ia es, makes i aluable in gene al
clinical esea ch. The me hod can, o example, o ecas he maximal
accu acy ob ainable by any in e ence algo i hm based on he same
p edic o s o a subse o hose p edic o s; and i a ains, by cons uc ion,
ha maximal accu acy. Fu he ea u es o in e es in machine lea ning
a e discussed in he nex sec ion.
The so wa e o implemen ing Bayesian nonpa ame ic in e ence
is a he momen a ailable as a clinician- iendly p o o-package
6
in he
R p og amming language7.
The me hodology unde lying Bayesian nonpa ame ics has been
success ully demons a ed o Alzheime ’s Disease wi h a smalle numbe
o p edic o s
8
, is used in many applica ions in as ophysics
9
, and i s
5
see e.g. Edmonds e al. 2015;2020, whose me hods we ind, howe e , inconclusi e.
6h ps://pglpm.gi hub.io/in e no
; he sc ip s used in he p esen wo k a e a ailable
a doi:
10.17605/os .io/zb26
.
7
R Co e Team 2023.
8
An oniano-Villalobos e
al. 2014.9E en Ho izon Telescope Collabo a ion 2019;2022; Del Pozzo e al. 2018.
6
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
ad an ages in neu oc i ical ca e and medicine ha e been emphasized o
qui e some ime10.
The nex sec ion 2gi es an in ui i e unde s anding o he me hod’s
unde lying p inciples and wo kings. Conc e e applica ion o he me hod
is shown in § 3, using he ou -pa ien ic i ious scena io o § 1.0 as a
conc e e example, and subsec ion 3.4 discusses u he applica ions o
gene al medical esea ch. A summa y and discussion is gi en in § 4.
Ma hema ical de ails and p oo s on which he p esen wo k is g ounded
a e gi en in a companion echnical no e11.
We apologize o eade s who may ind some discussions o explana-
ions oo ob ious, o some ma hema ical de ails oo sca ce. We wan ed
he p esen wo k o be accessible o a wide audience, om clinicians and
s uden s o medicine o esea che s in machine lea ning and p obabili y
heo y.
2 Bayesian nonpa ame ic in e ence
This sec ion can be especially o in e es o eade s om Machine Lea ning
and A i icial In elligence. I is la gely independen o he nex one, which
desc ibes he me hod’s applica ion. I can be ead a e § 3by eade s
who would like o see he me hod in ac ion i s .
2.1 Unde lying heo y and cha ac e is ics
The me hod o sol e clinical decision-making p oblems such as he one
o § 1is none o he han Decision Theo y: he combina ion o p obab-
ili y heo y and u ili y heo y. I in eg a es a ailable clinical s a is ical
da a wi h each pa ien ’s unique combina ion o clinical esul s, auxili-
a y in o ma ion, and ea men bene i s, in a ma hema ical amewo k,
comple ely de e mined by basic sel -consis ency equi emen s.12
Medicine has he dis inc ion o ha ing been one o he i s ields
o adop Decision Theo y, wi h he pionee ing wo k by Ledley – who,
10
Jawa & Maslo e 2023; Temp e al. 2021; An oniano-Villalobos e al. 2014; Side-
bo ham 2020; Goodman 1999.
11 h ps://gi hub.com/pglpm/in e no/ aw/main/de
elopmen /manual/op imal_p edic o _machine.pd 12
Jaynes 2003 chs 13–14; on
Neumann & Mo gens e n 1955; Cox 1946; Sa age 1972; Luce & Rai a 1957; Rai a &
Schlai e 2000; Rai a 1970; Lindley 1988; K eps 1988.
7
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
inciden ally, died o Alzheime ’s Disease
13
– and Lus ed
14
, who also
p omo ed i s algo i hmic implemen a ion
15
. Clinical decision-making
is oday explained and exempli ied in b illian ex books o medical
s uden s and clinicians16. An ou line is gi en in § 3.3.
So wa e o Bayesian nonpa ame ics p o ides an algo i hmic imple-
men a ion, as d eamed by Lus ed and Ledley
17
, o he main calcula ions
unde lying he clinical decision-making p ocess: om he compa ison
o a pa ien ’s speci ic p edic o s wi h he s a is ics o e ed by a clinical
da abase, o he choice o op imal ea men .18
Decision heo y is also he no ma i e ounda ion o he cons uc ion
o an A i icial In elligence agen capable o a ional in e ence and
decision making
19
. Bayesian nonpa ame ics can he e o e be seen as
an ideal machine-lea ning algo i hm. I is “ideal” in he sense o being
ee om special modelling assump ions ( his is why we do no call i
a “model”) and om limi a ions o in o ma ional ou pu which a ec
mos common machine-lea ning algo i hms; no “ideal” in he sense o
being imp ac icable. Qui e he opposi e, he p esen wo k shows ha
his ideal machine-lea ning algo i hm can oday be used in a wide ange
o in e ence p oblems a insubs an ial compu a ional cos .
Mo e conc e ely, Bayesian nonpa ame ics is ideal because i compu es
he p obabili y dis ibu ion o e all possible long- un equency dis ibu ions
om which he lea ning da ase can o igina e, hese equency dis ibu ions
being join ones o all p edic o and p edic and a ia es. This is he
maximum possible amoun o in o ma ion ha can be ex ac ed om he
lea ning da ase , in a s ic in o ma ion- heo e ic sense. F om his p ob-
abili y dis ibu ion, Bayesian nonpa ame ics can indeed calcula e any
quan i y ou pu ed by o he machine-lea ning algo i hms. Fo example
20
:
•
“Disc imina i e” algo i hms: he p obabili y
p(𝑌|𝑋)
o any se o
p edic ands 𝑌gi en any se o inpu p edic o s 𝑋.
•
“Gene a i e” algo i hms: he p obabili y
p(𝑋|𝑌)
o any se o inpu
p edic o s 𝑋gi en any se o p edic and alues 𝑌.
13
Shah e al. 2013.
14
Ledley & Lus ed 1959a,b;1960; Lus ed & Ledley 1960; Lus ed 1967.
15
Lus ed 1968; Ledley 1959;1960 § 1-5 p. 21.
16
Weins ein & Finebe g 1980; Sox e al. 2013;
Hunink e al. 2014.
17
c . he Appendices in Lus ed 1968.
18
In p e ious d a s we
called his so wa e implemen a ion a “Lus ed-Jaynes machine” as a homage o Lus ed and
o Jaynes 2003, who b illian ly explained he induc i e logic unde lying such a “ obo ”.
19
Russell & No ig 2022 pa IV; Jaynes 2003 chs 1–2, 13–14.
20
o e minology see e.g.
Mu phy 2012 § 8.6.
8
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
Mo e gene ally, Bayesian nonpa ame ics can compu e any join ,
ma ginal, o condi ional p obabili ies
𝑝(𝑍′, 𝑍′′)
,
𝑝(𝑍′)
,
𝑝(𝑍′|𝑍′′)
o any desi ed subse s o a ia es 𝑍′, 𝑍′′.
•
Reg ession o classi ica ion: he expec ed alue
E(𝑌|𝑋)
o any se o
a ia es
𝑌
, gi en any o he se o a ia es
𝑋
, including he pa icula
case o
𝑌
p edic and, and
𝑋
p edic o s. The unce ain y o a iabili y
a ound such an a e age is also au oma ically compu ed.
•
Func ional eg ession: i he p edic and
𝑌
o any o he a ia e o
in e es u ns ou o be a unc ion
𝑓
o a ia es
𝑋
, hen hei
condi ional p obabili y will be a del a dis ibu ion:
p(𝑌|𝑋)=δ[𝑌−
𝑓(𝑋)]
. Thus Bayesian nonpa ame ics always eco e s a unc ional
ela ionship i he e is one, as well as i s noise dis ibu ion.
Fu he mo e, his me hod also quan i ies he unce ain y o all ou pu s
abo e. Mo e p ecisely, i akes in o accoun how he s a is ical p ope ies
o he lea ning da ase could be di e en om hose o i s o iginal
popula ion, owing o sampling luc ua ions; and i can compu e how
much any o he ou pu s abo e would p obably change i mo e lea ning
da a we e collec ed.
In he nex sec ion we explain in ui i ely how Bayesian nonpa a-
me ics compu es he gene al p obabili y dis ibu ion o e long- un
equencies. A couple o special cha ac e is ics b ough abou by such
compu a ion can al eady be summa ized he e. Fi s , in con as o
machine-lea ning algo i hms such as neu al ne wo ks, andom o es s,
Gaussian p ocesses, suppo - ec o machines, o gene alized linea mod-
els, Bayesian nonpa ame ics does no assume he exis ence o a unc ion
(possibly con amina ed by a li le noise) om p edic o s o p edic ands.
This is a e y s ong assump ion, jus i iable in he p esence o in o ma ion-
ally e y ich p edic o s such as images, bu o he wise qui e un ealis ic
o many kinds o p edic o s conside ed in medicine, especially hose
ha a e mo e eadily a ailable and less in asi e and, he e o e, mo e
desi able. Second, in con as o algo i hms such as neu al ne wo ks, an-
dom o es s, suppo - ec o machines, logis ic eg ession, o gene alized
linea models, Bayesian nonpa ame ics does no do an op imiza ion
du ing he lea ning phase, sea ching o he minimum o some objec i e
unc ion. I does a ull hypo hesis-space su ey. The op imiza ion done
by mos machine-lea ning algo i hms is an app oxima e o m o his
su ey, based on he assump ion o hope ha he mos ele an po ion
9
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
and
GDS
a ia es a e in ege - alued, hippocampal olume and
Age
a e
con inuous, and
APOE4
and
Sex
a e bina y. The alues o one o wo o
hese p edic o s we e missing o 30 subjec s in he da ase .
The Bayesian-nonpa ame ics so wa e ook less han i e hou s
(on a 16-co e In el Co e i9-12900K CPU) o calcula e he p obabili y
dis ibu ion o he possible join popula ion- equency dis ibu ions o
he 13 a ia es.
Some esul s can al eady be isualized a e his in e ence. Figu e 3
shows, on he le , he in e ed dis ibu ions o
RAVLT-del
,
RAVLT-imm
,
GDS
, and hippocampal olume o he subpopula ion o pa ien s ha will
con e o Alzheime ’s Disease ( ed) and he subpopula ion ha will
emain wi h s able Mild Cogni i e Impai men (blue). On he igh , he
in e ed equency o con e sion in he ull popula ion is plo ed (g ey),
condi ional on he same p edic o s. The hin cu es a e 100 samples o
highly p obable popula ion- equency dis ibu ions; he hicke lines
a e hei means, which a e also he p edic i e condi ional p obabili ies.
The wo subpopula ions o pa ien s a e clea ly dis inc in he
RAVLT-
del
,
RAVLT-imm
,
HV
a ia es. These p edic o s can yield p obabili ies o
con e sion as high as 70% o as low as 10%. The wo subpopula ions a e
p ac ically indis inguishable in he
GDS
a ia e, which, he e o e, always
gi es e y unce ain p edic ions.
The lea ning da ase comp ises enough da a o g ea ly educe ou
unce ain y abou he popula ion dis ibu ions, as e iden om he e y
na ow sp ead o he cu es. In ac i leads o iden ical answe s, wi hin
nume ical-compu a ion e o , e en i we d as ically change he p io
illus a ed in ig. 1, o example a ou ing mo e unimodal dis ibu ions
o mo e mul imodal dis ibu ions.
These simple esul s show he g ea use ulness o Bayesian nonpa a-
me ic so wa e o gene al medical esea ch.
3.1 Pa ien ’s clinical in o ma ion
The 12 p edic o alues o ou ou pa ien s a e epo ed in able 2, op.
No e ha Cu is’s alue o he Hippocampal Volume is missing; his
is no a p oblem o Bayesian nonpa ame ics. Gi en hese p edic o
alues he me hod can ou pu any p obabili ies o in e es o he clinician.
16
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
0 2 4 6 8 10 12 14
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
RAVLT−del
popula ion equency
will con e o AD
da a his og am
s able MCI
da a his og am
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RAVLT−del
popula ion equency o con e sion o AD
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
RAVLT−imm
popula ion equency
will con e o AD
da a his og am
s able MCI
da a his og am
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RAVLT−imm
popula ion equency o con e sion o AD
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
GDS
popula ion equency
will con e o AD
da a his og am
s able MCI
da a his og am
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
GDS
popula ion equency o con e sion o AD
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
HV
popula ion equency densi y
will con e o AD
da a his og am
s able MCI
da a his og am
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
HV
popula ion equency o con e sion o AD
Figu e 3 In e ed dis ibu ions o some p edic o a ia es, o he subpopula ion o
pa ien s ha will con e o Alzheime ’s Disease ( ed dashed) and he subpopula ion wi h
s able Mild Cogni i e Impai men (solid blue).
17
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
Oli ia A iel Bianca Cu is
Age 75.4 75.4 75.4 63.8
Sex F F F M
HV/10−34.26 4.26 4.26 [missing]
APOE4 N N N Y
ANART 18 18 18 15
CFT 21 21 21 14
GDS 3 3 3 2
RAVLT-imm 36 36 36 20
RAVLT-del 5 5 5 0
RAVLT- ec 10 10 10 3
TMTA 21 21 21 36
TMTB 114 114 114 126
p(cAD=Y|p edic o s)0.302 0.302 0.302 0.703
p(p edic o s |cAD=Y)/10−12 8.97 8.97 8.97 1.14
p(p edic o s |cAD=N)/10−12 18.6 18.6 18.6 0.343
Table 2 P edic o alues o he ou pa ien s (see § 3.0), and esul ing condi ional
p obabili ies.
Table 2, bo om, epo s h ee p obabili ies ha a e impo an o he s ep
o he nex subsec ion:35
•p(cAD=Y|p edic o s)
: he p obabili y ha he pa ien will con e
o Alzheime ’s Disease, gi en he pa ien ’s speci ic p edic o s and
ha he pa ien comes om he same popula ion as he lea ning
da ase .
•p(p edic o s |cAD=Y)
: he p obabili y ha a pa ien who will
con e o Alzheime ’s Disease would ha e hese speci ic p edic o
alues. In o he wo ds, he likelihood
36
o con e sion o Alzheime ’s
Disease, gi en he p edic o s.
•p(p edic o s |cAD=N)
: he p obabili y ha a pa ien who will
emain wi h s able Mild Cogni i e Impai men would ha e hese
speci ic p edic o alues. In o he wo ds, he likelihood o s able Mild
Cogni i e Impai men , gi en he p edic o s.
35
All ela i e unce ain ies o he esul s caused by nume ical compu a ion e o a e below
0.8%, Cu is’s wo likelihoods being an excep ion a 2%.
18
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
0
50
100
150
200
250
300
350
400
450
500
550
600
650
Cu is's HV
p obabili y densi y
Cu is
Whole popula ion
Figu e 4 P obabili y dis ibu ion o
Cu is’s Hippocampal Volume (g een). The
ull-popula ion dis ibu ion (dashed g ey) is
also plo ed o e e ence.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
equency o con e sion o AD, gi en p edic o alues
p obabili y densi y
Figu e 5 P obabili y dis ibu ion o he
equency o con e sion o ad in he subpop-
ula ion ha ing Oli ia’s p edic o s. The ed
e ical line is he alue o he p obabili y
p(cAD=Y|p edic o s).
Bayesian nonpa ame ics can also answe s o he ques ions o in e es
o he clinician. Fo ins ance, wha could be he alue o Cu is’s Hip-
pocampal Volume? The answe is gi en in ig. 4, which also shows
he ull-popula ion dis ibu ion as compa ison (dashed g ey); wi h 95%
p obabili y Cu is’s alue is be ween 2.8 and 5.3, wi h a median o 3.8.
And wha is he equency o con e sion o Alzheime ’s Disease among
he subpopula ion ha ing Oli ia’s, A iel’s, o Bianca’s p edic o s? The an-
swe is gi en in he his og am o ig. 5: wi h 95% p obabili y, he ac ion
o his subpopula ion ha e en ually con e s o Alzheime ’s Disease is
be ween 0.19 and 0.43; his unce ain y ange is due o he limi ed size
o he lea ning da ase . The p obabili y
p(cAD=Y|p edic o s)
is equal
o he a e age o such a dis ibu ion
37
, p o ided he pa ien and da ase
can be conside ed as belonging o he same popula ion.
3.2 Assessmen o ele an subpopula ion and auxilia y in o ma ion
Ra ionale
As al eady men ioned, and as will be a gued mo e conc e ely in he nex
sec ion, he clinician needs a p obabili y in o de o choose a ea men
o o he cou se o ac ion o he cu en pa ien . This p obabili y is com-
pu ed by gene alizing associa ions be ween p edic o s and p edic and
36 p(𝐴|𝐵)
is he p obabili y o
𝐴
gi en
𝐵
, as well as he likelihood o
𝐵
gi en
𝐴
(Good 1950
§ 6.1). 37 e.g. Be na do & Smi h 2000 §§ 4.2–4.3.
19
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
hidden in a da ase o simila pa ien s, as discussed in § 2. The way his
gene aliza ion is made, howe e , can di e om pa ien o pa ien in wo
espec s:
•
Only some pa icula di ec ed associa ions can be gene alized o
he cu en pa ien , whe eas o he s would be inapp op ia e o
gene alize. In some cases, o example when he lea ning da ase
is a i icially assembled wi h balancing o s a i ica ion me hods,
some associa ions canno be gene alized o any pa ien s a all.
•
The e can be addi ional in o ma ion a ailable o he cu en pa ien ,
o ins ance some clinical p edic o s no included in he lea ning
da ase , o o he “so e ” in o ma ion such as amily his o y o
geog aphic backg ound.
The e is no sha p sepa a ion be ween hese wo i ems. The p esence o
addi ional in o ma ion o en au oma ically implies ha some associa ions
canno be gene alized om he lea ning da ase o he cu en pa ien .
Le us explain wi h a amilia example why
pa icula associa ions canno be gene alized:
he base- a e allacy
38
. Conside a la ge se o clin-
ical ials, illus a ed in he uppe able on he
side, whe e each do ep esen s, say, 10 000 ial
pa ien s. In his sample da ase i is ound ha ,
among pa ien s ha ing a pa icula alue “+”
o some p edic o s (le column),
5/7=71.4%
o hem (uppe squa e) e en ually de eloped a
disease. The allacy lies in judging ha a new
pa ien om he ull popula ion, who also has
p edic o alue “+”, also has a 71.4% p obabili y
o de eloping ha disease. In ac , his p obabili y
will in gene al be di e en . In ou example, i is
5/15 =33.3%
, as can be seen in he lowe able
illus a ing he ull popula ion. This di e ence
would be no iced as soon as he inapp op ia e p obabili y was used o
make p ognoses in he ull popula ion. A simila si ua ion happens o
he o he p edic o alue “−”.
The disc epancy in he condi ional equencies o p edic and gi en
p edic o s, be ween he sample da ase and he ull popula ion, a ises
38 Ba -Hillel 1980; Jenny e al. 2018; Sp enge & Weinbe ge 2021; Ma hews 1996.
20
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
because he p opo ion o posi i e s nega i e disease cases in he
la e has some alue, 16.7%/83.3% in ou example, whe eas he ial
samples (dashed line in he lowe able) we e hand-chosen so as o
ha e a 50%/50% p opo ion. This sampling p ocedu e is called “class
balancing” in machine lea ning
39
. Mo e gene ally his disc epancy can
appea whene e a popula ion and a sample do no ha e he same
equency dis ibu ion o he p edic and. In his case we canno ely
on he p obabili ies o “p edic and gi en p edic o s” ob ained om he
sample da ase , which we symbolically w i e as
p(p edic and |p edic o s,da ase )(1)
A li le coun ing in he side igu e e eals, howe e , ha o he e-
quencies may be elied upon. Conside he ull popula ion. Among all
pa ien s who de eloped he disease,
5/6=83.3%
o hem (uppe ow)
had p edic o alue “+”, while among hose who did no de elop he
disease,
2/6=33.3%
(lowe ow) had p edic o alue “
−
”. And hese
equencies a e he same in he sample da ase . These equencies om he
clinical ials can he e o e be used o make a p ognosis abou he ull
popula ion, using Bayes’s heo em. Fo b e i y, deno e he p edic o s by
𝑋
, he p edic and by
𝑌
, he da ase o ials by
𝐷
, and he ull-popula ion
base a e by 𝑅. Bayes’s heo em yields
p(𝑌|𝑋, 𝐷, 𝑅)=
p(𝑋|𝑌, 𝐷) · p(𝑌|𝑅)
P
𝑌
p(𝑋|𝑌, 𝐷) · p(𝑌|𝑅)(2)
In ou example we ind
p(𝑌=Y|𝑋=+, 𝐷, 𝑅)
=
p(𝑋=+ | 𝑌=Y, 𝐷)·p(𝑌=Y|𝑅)
p(𝑋=+ | 𝑌=Y, 𝐷)·p(𝑌=Y|𝑅) + p(𝑋=+ | 𝑌=N, 𝐷)·p(𝑌=N|𝑅)
≈0.833 ·0.167
0.833 ·0.167 +0.333 ·0.833
=0.33
(3)
which is indeed he co ec ull-popula ion equency.
I he samples o he clinical ials had been chosen wi h he
same equencies as he ull popula ion (no “class balancing”), hen
39 P o os 2000; D ummond & Hol e 2005; Weiss & P o os 2003.
21
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
he p obabili y
p(p edic and |p edic o s,da ase )
om he da a-
se would be he app op ia e one o use. Bu he p obabili ies
p(p edic o s |p edic and,da ase )
oge he wi h Bayes’s heo em
as in eq.
(2)
would also lead o exac ly he same p obabili y. We hus see
ha using he p obabili ies
p(p edic o s |p edic and,da ase )
om he da ase is p e e able o using
p(p edic and |p edic o s,da ase )
.
The o me yield he same esul s as he la e when use o he la e is ap-
p op ia e, and allow us o apply co ec ions when use o he la e is inap-
p op ia e. The supe io i y o using
p(p edic o s|p edic and,da ase )
p obabili ies (called “gene a i e” in machine lea ning, see e.g.
40
) is illus-
a ed wi h a oy example in able 3.
The use o da ase p obabili ies di e en om
p(p edic and |p edic o s,da ase )
can be necessa y e en when he da ase has s a is ics iden ical wi h
he popula ion i is sampled om. Typical cases a e he p ognosis o a
pa ien ha comes om a peculia subpopula ion o e en om a di e en
popula ion
41
. Fo ins ance, he i s case happens when he clinician has
addi ional in o ma ion no included among he p edic o a ia es, such
as he esul o an addi ional clinical es , o amily his o y; he second
case happens when he pa ien comes om a di e en geog aphical
egion. The e is o cou se no sha p dis inc ion be ween hese wo cases.
Wha is impo an is ha , in ei he case, i can s ill be possible
o use s a is ical in o ma ion om he sample da ase o make p o-
gnoses. I is su icien ha some condi ional s a is ics may be applic-
able o he speci ic pa ien . Fo a pa ien coming om a di e en e-
gion, o example, i may be judged ha he condi ional p obabili ies
p(p edic and |p edic o s,da ase )
s ill apply. In o he wo ds, he pa-
ien may s ill be conside ed a membe o he subpopula ion ha ing hose
speci ic p edic o alues. Using mo e echnical language we say ha a
new pa ien can be conside ed exchangeable wi h he pa ien s cons i u ing
he da ase , bu only condi ional on pa icula a ia es. Fo a clea and
logically impeccable p esen a ion no obscu ed by echnical language,
40
Mu phy 2012 § 8.6.
41
Lindley & No ick 1981; Quin ana e al. 2017; Sox e al. 2013
ch. 4; Hunink e al. 2014 ch. 5.
22
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
Table 3 Supe io i y o he “p edic o s |
|
|p edic and” (o “gene a i e”) app oach
We spli ou lea ning da ase in o wo subse s:
•
One wi h 361 subjec s and a a io o 29.9%/70.1% o subjec s wi h
cAD=Y s cAD=N.
•
One wi h 343 subjec s and a a io o 63.3%/36.7% o subjec s wi h
cAD=Y s cAD=N. This subse is used as a ic i e ull popula ion.
This pa i ion was made wi h no sys ema ic sampling o any a ia es excep
he p edic and cAD.
A e aining on he lea ning da ase , we make a p ognosis o each o he
343 “new” pa ien s, h ough ou sepa a e app oaches: (a) using he p obab-
ili ies
p(p edic and |p edic o s,da ase )
, as ypical o machine-lea ning
algo i hms; (b) using
p(p edic o s |p edic and,da ase )
oge he wi h
he base a e, as explained abo e; (c) ossing a coin; (d) always p ognosing
“
cAD=Y
”, which gua an ees 63.3% co ec p ognoses owing o he base a e
o he ull popula ion. Finally, he accu acies (numbe o p ognoses gi ing
mo e han 50% p obabili y o he co ec ou come) o hese ou app oaches
a e calcula ed. He e a e he esul s om lowes o highes :
p edic and |p edic o s 37.3%
coin oss 50%
always p edic con e sion 63.3%
p edic o s |p edic and & base a e 73.2%
The “p edic and
|
p edic o s” app oach (“disc imina i e” in machine-
lea ning pa lance) leads o wo se esul s han a coin oss because o i s
unde lying base- a e allacy. The “p edic o s
|
p edic and” app oach (“gen-
e a i e” in machine-lea ning pa lance) leads o be e esul s han simply
always p ognosing he mos common base- a e ou come; his shows ha he
da ase can s ill p o ide use ul s a is ical in o ma ion despi e i s misma ched
base a e. In e ence algo i hms ha only yield “p edic and
|
p edic o s” ou -
pu s, unlike Bayesian nonpa ame ics, a e incapable o ex ac ing his use ul
in o ma ion.
23
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
see Lindley
42
. This opic is also igh ly ela ed o con ounding and o
Simpson’s pa adox43.
This opic is complex and o ex eme impo ance o in e ence, bu
i s de ailed s udy is no he goal o he p esen wo k. Ou main poin
he e is ha popula ion a iabili y and auxilia y clinical in o ma ion a e
impo an ac o s ha di e en ia e pa ien s, and a pe sonalized app oach
ough o ake hem in o accoun . The me hod he e p esen ed does
his na u ally, allowing a g ea lexibili y in selec ing which s a is ical
ea u es o he sample da ase should be used o each new pa ien ,
and he in eg a ion o auxilia y clinical in o ma ion in he o m o a
p io p obabili y. As discussed in § 3.1, Bayesian nonpa ame ics allows
us o quickly calcula e condi ional p obabili ies
p(𝑌|𝑋, da ase )
o
any desi ed a ia e subse s
𝑌
and
𝑋
equi ed by he pa ien ’s ele an
popula ion.
Applica ion o he example s udy
In ou example o § 1.0, all s a is ics o he da ase a e conside ed ele an
o Oli ia, Bianca, and Cu is. Fo hese pa ien s he clinician can he e o e
use Bayes’s heo em wi h he likelihoods o able 2and he da ase
con e sion a e o
0.463
– o equi alen ly di ec ly he p obabili ies
p(cAD=Y|p edic o s,da ase )p o ided in he same able.
Fo A iel, howe e , he clinician judges ha a di e en base a e o
p io p obabili y o con e sion should be used, equal o 65%, because
o he di e en geog aphical o igin and amily his o y. In he case he
clinician uses Bayes’s heo em wi h he likelihoods o able 2and he
p io p obabili y o 0.65.
The inal p obabili ies o con e sion o Alzheime ’s Disease o ou
ou pa ien s a e epo ed in able 4. No e how he inal p obabili y o
A iel is highe han ha o Oli ia and Bianca, e en i he p edic o da a
a e he same o hese h ee pa ien s.
42
Lindley 2014 especially a ound §§ 7.3, 8.6; Lindley & No ick 1981; mo e echnical
e e ences a e de Fine i 1930;1937; Dawid 2013; Be na do & Smi h 2000 §§ 4.2–4.3, 4.6.
43 Malinas & Bigelow 2016; Sp enge & Weinbe ge 2021.
24
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
3.3 Assessmen s o ea men s and bene i s; inal decision
Ra ionale
A c ucial poin in clinical decision-making is his: he clinician needs o
assess, no he p esence (p esen o u u e) o a disease, bu he isk o i s
p esence. Is he e a di e ence be ween hese wo p oblems? and why is
he di e ence impo an ?
In clinical p ac ice, we can a ely diagnose o p ognose a medical
condi ion wi h ull ce ain y. Pe ec classi ica ion is he e o e impossible.
Bu also a “mos p obable” classi ica ion, which may be enough in
o he con ex s, is inadequa e in clinical ones. The p oblem is ha he
clinician has o decide among di e en cou ses o ac ion, such as di e en
ea men s, mo e es s, and so on, and he op imal one depends on how
p obable he medical condi ion is, no jus on whe he i is mo e p obable
han no .
Two examples illus a e his poin . Suppose he e is a dange ous
ea men ha ex ends he pa ien ’s li e ime by 1 yea i he disease is on
i s cou se, bu sho ens he pa ien ’s li e ime by 5 yea s i he disease is
no p esen . Also suppose ha some algo i hm ells he clinician whe he
he disease’s p esence is “mo e p obable han no ”, gi en some p edic o
alues; in which case he clinician adminis e s he dange ous ea men .
I u ns ou ha 60 ou o 100 ea ed pa ien s ha ing hese same p edic o
alues e en ually de elop he disease, so “mo e p obable han no ” is
co ec . Howe e , he inal esul is ha he clinician has added
1×60 =60
yea s bu also sub ac ed
5×40 =240
yea s om he combined li espans
o he ea ed pa ien s! The conclusion is ha he ea men canno be
p esc ibed jus because he disease is “mo e p obably p esen han no ”.
As an opposi e example, suppose ha a less dange ous ea men ex ends
he pa ien ’s li espan by i e yea s i he disease is on i s cou se, bu
sho ens i by one mon h i he disease is no p esen . In his case, i may
Oli ia A iel Bianca Cu is
ini ial p obabili y
p(cAD=Y|aux in o)0.463 0.65 0.463 0.463
inal p obabili y
p(cAD=Y|p edic o s,da ase ,aux in o)0.302 0.47 0.302 0.703
Table 4 Final p obabili ies o con e sion compu ed om da ase and auxilia y in o ma ion
25
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
( he ou co esponding p obabili y his og ams, i plo ed join ly, would
look like dis inc e ical lines). I is clea ha knowledge o he Hippo-
campal Volume is ex emely unlikely o change Cu is’s op imal ea men
om
𝛿
. Conside ing ha he negligible in o ma ion gained would no
ou weigh he economic cos s (in ol ing an m i-scan) o ob aining his
p edic o , he clinician decides o p oceed wi hou i .
P edic o impo ance
The ques ion abou Cu is in he p e ious subsec ion can be gene alized
o a whole popula ion. P edic o s ha a e oo in asi e o oo expensi e
o ob ain, bu ha a e unin o ma i e o he p ognosis, could be d opped
al oge he . So how impo an , in gene al, is each p edic o in p ognosing
he con e sion o Alzheime ’s Disease?
As posed, his ques ion is oo ague (ill-posed) because i does no
exac ly speci y how a p edic o is used, and wha “impo an ” means.
Le us see why hese de ails ma e .
X1
X2
The schema ic pic u e on he side illus a es
he necessi y o speci ying a p edic o ’s con ex .
Indi iduals in his popula ion can be ei he
blue ci cles
o ed iangles
△
, and ha e wo
p edic o s
𝑋1
and
𝑋2
. P edic o
𝑋1
, i used by
i sel , is wo hless in dis inguishing he wo
subpopula ions, because hese ha e iden ical
ma ginal dis ibu ions (depic ed unde nea h
he g ey ho izon al line). I used in conjunc ion wi h
𝑋2
, howe e ,
p edic o
𝑋1
allows us o iden i y an indi idual’s subpopula ion wi h
ull ce ain y, as is clea om he wo-dimensional iew. I is he e o e
an essen ial p edic o in his case: d opping i would lead o a comple e
loss o p edic i e powe . An analogous discussion holds o
𝑋2
in he
p esen case. The con e se can also happen (no illus a ed): a p edic o
migh be “good” i used by i sel , and ye i migh be disca ded wi hou
any loss i used in combina ion wi h o he s.
In ou ques ion abou a p edic o ’s impo ance, we wan o know
wha happens i he p edic o is d opped om he se o all p edic o s.
Rega ding he meaning o “impo ance” o “p ognos ic powe ”, we
mus speci y a ele an me ic, and p edic o s could be anked di e en ly
by di e en me ics. F om ou discussion so a i is clea ha in clinical
32
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
decision-making he canonical me ic is he inal expec ed u ili y – and
he e o e he choice o op imal ea men – which a p edic o ’s p esence
o absence leads o (see § 3.3). This poin was illus a ed wi h Cu is’s
example in he p e ious subsec ion. Wha i we wan o make a simila
assessmen , no o a single pa ien , bu o he ull popula ion? which
u ili y ma ix should we use? I can be p o ed, again om decision-
heo e ic p inciples, ha he popula ion a e age o all u ili y ma ices
should be used in his case
49
. This seems a quan i y e y di icul o assess,
bu i can also be shown
50
ha e en a semi-quan i a i e assessmen leads
o be e esul s han using some o he gene al-pu pose me ic.
Bayesian nonpa ame ics allows us o compu e he expec ed alue
o i ually any p ognos ic-impo ance me ic, and o any subse o
p edic o s a ailable in he da ase . This compu a ion has mo eo e wo
p ope ies o pa amoun impo ance: (a) he p ognos ic powe o a se
o p edic o s ound wi h Bayesian nonpa ame ics is he maximum possible
ob ainable by any in e ence algo i hm, o in o he wo ds i is an in insic
p ope y o ha se o p edic o s; (b) Bayesian nonpa ame ics achie es his
maximum powe . Thus, i Bayesian nonpa ame ics says ha he accu acy
ob ainable wi h a gi en se o p edic o s is 70%, hen we know ha no
o he in e ence algo i hm can each a highe accu acy han 70%; in e ence
algo i hms ha each lowe accu acy can in p inciple be imp o ed upon.
Bayesian nonpa ame ics, by cons uc ion, will each his accu acy. No e
ha we mean accu acy in he long un, o e he ull popula ion; an
in e ence algo i hm could each highe accu acies in some es da ase
hanks o sampling luc ua ions; in ac his is bound o happen om
ime o ime.51
Le us illus a e his kind o “p edic o impo ance” assessmen
o ou da ase . We use (a) wo me ics: he accu acy and he mu ual
in o ma ion
52
be ween a se o p edic o s and he
cAD
p edic and; (b) 27
di e en se s o p edic o s:
•e e y p edic o , used indi idually (12 se s);
•
all cogni i e- es p edic o s used oge he , join ly wi h in o ma ion
abou dep ession (GDS) and demog aphics (Age and Sex).
49
c . Dy land e al. 2022a § 4.1.
50
Dy land e al. 2022a § 4.2.
51
Bayesian nonpa ame ics
can also calcula e, wi h a somewha expensi e compu a ion, he size o such luc ua ions,
gi en he size o he es da ase . 52 Shannon 1948; Co e & Thomas 2006.
33
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
•APOE4
and Hippocampal Volume, join ly wi h demog aphic in o m-
a ion;
•
all p edic o s join ly excluding one, each single p edic o being
excluded in u n (12 se s);
•all p edic o s join ly.
Use o he accu acy assumes ha he popula ion o pa ien s has only
wo a ailable ea men s ha ing a e age u ili y ma ix
1 0
0 1
. Mu ual
in o ma ion is a model- ee measu e o he ela ion be ween wo se s
o a ia es, wi h di e se ope a ional in e p e a ions
53
and in e na ional
s anda ds
54
. A se o p edic o s and a bina y a ia e (such as ou con e -
sion o Alzheime ’s Disease) ha e a mu ual in o ma ion o
1 Sh
i and
only i he e is a non-cons an de e minis ic unc ion om he o me o
he la e .
Ou speci ic ques ions a e he ollowing: “Wha is he expec ed alue
o he accu acy o he nex new pa ien , i we use he gi en se o
p edic o s?” and “Wha is he mu ual in o ma ion be ween he gi en se
o p edic o s and he p edic and, gi en he p esen ly a ailable da a?”.
The answe s o hese ques ions a e epo ed in ig. 7, o de ed om
bo om o op acco ding o inc easing me ic. The o de ing o mu ual
in o ma ion and accu acy ag ee wi hin he unce ain y o he nume ical
compu a ion (Mon e Ca lo in eg a ion). The la e is epo ed as co e age
in e als o ± wo s anda d de ia ions.
The plo s e eal se e al indings, alid wi hin he popula ion selec ed o
he da ase , which can be compa ed wi h he analysis in Rye e al.55:
•
The se o 12 p edic o s conside ed in he p esen wo k and in Rye e
al.
56
can a mos yield a p ognos ic accu acy o a ound
67.7%±0.7%
o e he ull popula ion, o any in e ence algo i hm. This ac
ag ees wi h he (comple ely independen ) indings in Rye e al.
57
,
whe e a maximal accu acy o 68.3% on a es da ase was ound
using an ensemble model. The p esen analysis also shows ha he
ensemble model managed o achie e he maximal accu acy possible
wi h hese p edic o s (bu see § 4 o limi a ions o ha model).
•
The mu ual in o ma ion using all 12 p edic o s is qui e low a
(0.140 ±0.008)Sh
, indica ing ha we canno easonably conside
53
MacKay 2005; Woodwa d 1964; Minka 2003; Good 1961; Good & Toulmin 1968;
Kelly 1956; Kullback 1978.
54
iso 2008.
55
Rye e al. 2022 see especially Fig. 3 and Table 3.
56 Rye e al. 2022.57 Rye e al. 2022.
34
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78
accu acy (expec ed u ili y)
APOE4
Sex
GDS
Age
ANART
TMTA
CFT
HC
TMTB
APOE4+HC+Age+Sex
RAVLT− ec
RAVLT−imm
RAVLT−del
all minus RAVLT−del
cogni i e+Age+Sex
all minus RAVLT−imm
all minus HC
all minus GDS
all minus Age
all minus APOE4
all minus TMTA
all minus Sex
all minus CFT
all minus ANART
all minus RAVLT− ec
all minus TMTB
all
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22
mu ual in o ma ion/Sh
GDS
Sex
APOE4
Age
ANART
TMTA
TMTB
CFT
HC
APOE4+HC+Age+Sex
RAVLT− ec
RAVLT−imm
RAVLT−del
all minus RAVLT−del
all minus RAVLT−imm
all minus TMTB
all minus RAVLT− ec
all minus HC
cogni i e+Age+Sex
all minus TMTA
all minus CFT
all minus Age
all minus GDS
all minus ANART
all minus Sex
all minus APOE4
all
Figu e 7 Expec ed accu acy o he nex new pa ien (le ), and mu ual in o ma ion ( igh ),
o se e al se s o p edic o s o he p ognosis o con e sion o Alzheime ’s Disease. Each
g aph has been e ically o de ed acco ding o inc easing alues; he wo ankings ag ee
wi hin he espec i e unce ain ies. The
all
p edic o se is ma hema ically gua an eed o
be op imal acco ding o bo h me ics and has he e o e been anked i s . Ba s show he
unce ain y in e al (± wo s anda d de ia ions).
he p edic and o be an app oxima e unc ion o he p edic o s (
0 Sh
co esponds o a coin oss,
1 Sh
o a pe ec unc ion). Machine-
lea ning algo i hms based on unc ional eg ession, such as neu al
ne wo ks, a e he e o e no app op ia e o his p ognos ic p oblem.
•APOE4
,
GDS
,
Age
,
Sex
, and o some deg ee
ANART
a e poo p edic o s
(wi hin his popula ion) when used alone and when used in com-
bina ion wi h all o he p edic o s. The la e poin is e iden om
he ac ha he mu ual in o ma ion and accu acy o he combined
p edic o s ba ely dec eases i any one o hese ou p edic o s is
omi ed.
•
The combined cogni i e and demog aphic a ia es a e be e p e-
dic o s han he join use o Hippocampal Volume,
APOE4
, and
demog aphic a ia es.
35
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cu is's unknown HV
Cu is's p obabili y o con e sion o AD
Figu e 8 P obabili y o
cAD=Y
o Cu is, gi en Cu is’s known p edic o s and di e en
possible alues o his unknown Hippocampal Volume. The hinne cu es a e 100 p obable
samples o how his p obabili y would change wi h a la ge lea ning da ase . Compa e
his igu e wi h ig. 3,p.17, bo om- igh .
•RAVLT-imm
,
RAVLT-del
, and o a lesse deg ee
RAVLT- ec
a e good
p edic o s, bo h when used alone and when used join ly wi h all
o he p edic o s. Hippocampal Volume is a poo e p edic o han
any o he
RAVLT
when used alone, and likely also when used
in combina ion wi h all o he s
58
. This las inding is also clea in
Cu is’s case: ig. 8shows ha his p obabili y o con e sion o
Alzheime ’s Disease, gi en his cu en p edic o s, would p ac ically
be he same o all alues o Hippocampal Volume; and i would
p obably be he same e en i he lea ning da ase con ained mo e
poin s.
Bayesian nonpa ame ics shows ha he omission o any one o he
12 p edic o s, excep
RAVLT-del
and possibly
RAVLT-imm
, does no lead
o an app eciable dec ease in accu acy ( ela i e dec ease o 0.3% o less)
o in mu ual in o ma ion ( ela i e dec ease o less han 3%). This pu s
he p ognos ic-impo ance analysis o Rye e al.
59
in o pe spec i e. The
58 con as his wi h Rye e al. 2022.59 Rye e al. 2022.
36
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
exac quan i ica ion o hese sub le di e ences is compu a ionally qui e
expensi e, and we did no ca y i ou u he .
4 Discussion
Which equi emen s does a pe sonalized app oach o p ognosis and
ea men impose on assis i e compu a ional echnology? This is an
impo an ques ion, because wi h he inc easing amoun o s a is ical
clinical da a and clinical p edic o s a ailable o medical ca e, assis i e
compu a ional echnology is oday no me ely a use ul op ion, bu a
necessi y in clinical p ac ice.
In he p esen wo k we s a ed om he pe spec i e o he clinician’s
ul ima e ask, decision-making unde isk, and saw ha pa ien s’ di e ences
ele an o p ognosis and ea men can be app oxima ely di ided in o
h ee ca ego ies:
•
di e ences in he alues – and a ailabili y – o a co e se o clinical
p edic o s, o which we ha e popula ion-wide s a is ical in o ma-
ion;
•
di e ences in he a ailabili y and alues o auxilia y and usually
semi-quan i a i e clinical in o ma ion, such as geog aphical o
amily backg ound;
•
di e ences in he a ailabili y and alues o “u ili ies” o clinical
cou ses o ac ion, such as p e en i e ea men s o u he es s;
such alues can ha e a highly a iable, pa ien -dependen subjec i e
componen .
Luckily he e is a heo y ha akes in o accoun and in eg a es hese
di e ences owa ds he inal goal: Decision Theo y, which is he subjec o
se e al good ex books on clinical decision-making
60
a e he pionee ing
wo k o Lesley & Lus ed 1959a,b;1960;1960;1968 (a summa y and
e e ences we e gi en in § 3.3).
Decision-making unde isk equi es any assis i e algo i hm o wo k,
explici ly o implici ly, in e ms o p obabili ies, ha ing p ecise con-
nec ions wi h popula ion s a is ics (§3.3). Wi hou his condi ion he
in eg a ion o pa ien -dependen ea men u ili ies would be impossible.
60 Weins ein & Finebe g 1980; Sox e al. 2013; Hunink e al. 2014.
37
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
Table 6 Summa y o he clinician’s pa ien -dependen inpu s and Bayesian-nonpa ame ics
ou pu s. Inpu da a and inal esul s ha dis inguish A iel, Bianca, Cu is om Oli ia a e
in ed.
Oli ia A iel Bianca Cu is
Clinician’s pa ien -dependen inpu s
P edic o alues
Age 75.4 75.4 75.4 63.8
Sex FFFM
HV/10−34.26 4.26 4.26 [missing]
APOE4 NNNY
ANART 18 18 18 15
CFT 21 21 21 14
GDS 3332
RAVLT-imm 36 36 36 20
RAVLT-del 5550
RAVLT- ec 10 10 10 3
TMTA 21 21 21 36
TMTB 114 114 114 126
Addi ional in o ma ion
auxilia y in o none amily his o y,
base a e none none
applicable subpopula ion all p edic o |p edic and all all
p io p obabili y
o con e sion 0.463 0.65 0.463 0.463
A ailable ea men s
and u ili ies
ea men 𝛼
ea men 𝛽
ea men 𝛾
ea men 𝛿
cAD
N Y
10 0
9 3
8 5
0 10
cAD
N Y
10 0
9 3
8 5
0 10
cAD
N Y
10 0
8 3
7 5
0 10
cAD
N Y
10 0
9 3
8 5
0 10
Bayesian-nonpa ame ics ou pu s
p(cAD=Y|p edic o s,da ase )0.302 0.302 0.302 0.703
p(p edic o s|cAD=Y,da ase )/10−12 8.97 8.97 8.97 1.14
p(p edic o s|cAD=N,da ase )/10−12 18.6 18.6 18.6 0.343
inal p obabili y o con e sion
p(cAD=Y|p edic o s,da ase ,aux in o)0.302 0.47 0.302 0.703
exp. u ili y ea men 𝛼
exp. u ili y ea men 𝛽
exp. u ili y ea men 𝛾
exp. u ili y ea men 𝛿
Op imal ea men
6.98
7.19
7.09
3.02
𝜷
5.27
6.16
6.58
4.73
𝜸
6.98
6.49
6.40
3.02
𝜶
2.97
4.78
5.89
7.03
𝜹
38
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
The handling o hese p obabili ies should mo eo e be enough lexible
o ake in o accoun peculia bu common subpopula ions o pa ien s
ha ing special con ex s o auxilia y in o ma ion (§ 3.2), and he com-
mon possibili y o missing alues o some clinical p edic o s (§3.1).
Mos , i no all, popula machine-lea ning algo i hms ei he do no mee
hese equi emen s, o hey do so a he cos o un ealis ic modelling
assump ions. Un o una ely hey end o o e ly simpli y he p oblem
o decision-making unde isk, as i i we e a simple classi ica ion o
eg ession ask.
We p esen ed an assis i e me hod and so wa e, implemen ing
Bayesian nonpa ame ic in e ence, ha mee s all hese equi emen s
(§ 2) and ca ies ou he calcula ions equi ed by decision heo y. This
me hod is mo eo e model- ee, no making a-p io i assump ions abou
unc ional dependencies o pa icula dis ibu ions in he a ia es. The
in e ence p inciples on which i is based ha e ecen ly been ecommen-
ded o he s udy o Alzheime ’s Disease
61
, and ha e been success ully
demons a ed in a simple p edic o se ing
62
. We showed i s applica ion
in an example o p ognosis and ea men o con e sion om Mild
Cogni i e Impai men o Alzheime ’s Disease o ou di e en pa ien s,
whe e all h ee ca ego ies o di e ences lis ed abo e appea ed. The
pa ien s we e ic i ious bu he unde lying lea ning da abase, o igina ing
om adni, is eal and was explo ed in a p e ious wo k63.
Bayesian nonpa ame ics was also shown o ha e uses ha go bey-
ond indi idual clinical decision-making bu a e s ill o impo ance o
pe sonalized medicine. Fo ins ance, i can assess he maximum possible
p ognos ic powe o pa icula se s o p edic o s, po en ially allowing us
o disca d clinical p edic o s ha a e oo in asi e o expensi e and ye
p ognos ically unimpo an .
In ac ual deploymen , we would ecommend he hospi al, medical
cen e, o clinician using Bayesian-nonpa ame ics so wa e o keep a
da abase o incoming pa ien s, wi h hei p edic o alues, adding he
ue alues o hei p edic and la e in ime, once hey become known.
The so wa e can hen be e ained on such local da abase when he la e
eaches a size compa able o he o iginal one’s, and pe iodically e ained
a e wa ds. All in e ences would hus become inc easingly mo e eliable,
61
Temp e al. 2021; see also asa 2016;2019; Jawa & Maslo e 2023; Sidebo ham 2020;
Goodman 1999.62 An oniano-Villalobos e al. 2014.63 Rye e al. 2022.
39
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
because he so wa e would base hem on upda ed popula ion s a is ics
ha a e cha ac e is ic o he speci ic hospi al.
4.1 Coun e s o possible c i iques
Any in e ence o decision-making algo i hm aspi ing o ake in o accoun
pa ien di e ences mus pe o ce ha e some open “inpu slo s” o such
di e ences. We saw ha Bayesian nonpa ame ics equi es inpu s abou
a pa ien ’s speci ic p edic o s, ele an s a is ical ela ions and auxilia y
da a, and ea men u ili ies.
The mos di icul inpu o quan i y is p obably he hi d: ansla ing
bene i s and d awbacks o di e en ea men s in o numbe s. On his
complex opic we e e he eade o specially dedica ed ex books on
clinical decision making, o example Sox e al. 2013 and Hunink e al.
2014.
Bu some eade s may wonde : “can all hese addi ional inpu s be
a oided?”, ea ing ha e o s could sneak in h ough hem.
This ques ion is answe ed by a ma hema ical heo em a he e y co e
o decision heo y
64
, which is oo seldom emphasized: Any decision we
make, ei he (A) comes explici ly o implici ly h ough some se o u ili ies
and maximiza ion o hei expec a ions, o (B) is logically inconsis en .
The e is no hi d al e na i e. Thus he choice is no be ween using u ili ies
o no using u ili ies, bu be ween choosing hem explici ly o le ing
hem be chosen in a way we do no know. I we use a decision-making
algo i hm ha does no ask us o u ili ies, hen he algo i hm is in e nally
supplying u ili ies no chosen by us (and p obably di o ced om ou
speci ic p oblem), o , wo se, is commi ing logical inconsis encies.
The i s ad an age o explici ly ope a ing h ough u ili ies, p obabil-
i ies, decision heo y, is ha we a e, a he e y leas , su e o no ac ing in
a sel -con adic o y way. The second ad an age is ha he u ili ies used
o a i e a a decision appea openly in on o us. We can analyse and
change hem i we ind hem inapp op ia e o a speci ic p oblem. I hey
a e hidden, i is mo e di icul o analyse which a e inapp op ia e and
how hey should be changed.
The ac ha an algo i hm wo ks acco ding o decision heo y is also
an assu ance o s i ing owa ds heo e ical op imali y. This poin has
64
Sa age 1972; Luce & Rai a 1957; Rai a & Schlai e 2000; A kinson e al. 1964;
Fe guson 1967; Lindley 1988;1977; K eps 1988; Be na do & Smi h 2000; P a e al. 1996;
Lindley 2014; Pe ig ew 2019.
40
Po a Mana e al. Pe sonalized p ognosis & ea men wi h Bayesian nonpa ame ics
e y sub le consequences. Conside a non-op imal algo i hm ha leads
o sa ing 85 000 pa ien s ou o 100 000. Gi en hese numbe s i migh be
deemed a success. Bu wha i a heo e ically op imal algo i hm leading
o 95 000 sa ed pa ien s is easible? Wha shall we say o he amilies o
he 10 000 pa ien s who could ha e been sa ed bu we en’ ?
4.2 Range o applica ion o Bayesian-nonpa ame ics so wa e
Fi s le us emphasize, e en i i is ob ious, ha he quali y o he esul s
ob ained wi h Bayesian nonpa ame ic in e ence depends on he quali y
o he lea ning da ase . Any peculia sampling biases (o nume ical
e o s) in he da ase ha a e unknown o he clinician will a ec he inal
esul s. This is o cou se ue o any in e ence algo i hm. Bu we saw
ha Bayesian nonpa ame ics allows he clinician o co ec o pa icula
sampling biases p esen in he da ase , i hey a e known.
The ange o applica ion o cu en Bayesian-nonpa ame ics so wa e
has wo kinds o bounds: compu a ional and heo e ical.
The ac ha Bayesian nonpa ame ics ex ac s all a ailable in o ma-
ion om he da ase makes i compu a ionally expensi e (see § 2). A
p esen i canno be used wi h high-dimensional p edic o s: i ou da ase
had included a p edic o such as a
128 ×128 ×128
g eyscale m i image,
he lea ning s age would ha e aken a ound 100 yea s. App oxima e
bu much as e algo i hms such as neu al ne wo ks and andom o es s
a e hus, a p esen , s ill he only op ions wi h such p edic o s. The e is,
howe e , he in e es ing possibili y o combining hese as algo i hms
oge he wi h Bayesian-nonpa ame ics so wa e, as a pos -p ocesso
o hei aw ou pu . The so wa e ex ac s use ul in o ma ion usually
hidden in hei ou pu a a low compu a ional cos
65
; his in o ma ion
can hen be used o clinical decision-making as illus a ed in he p esen
wo k.
The sole assump ion unde lying Bayesian nonpa ame ic in e ence
and i s p ac ical use wi h new pa ien s, is ha he la e can be assumed
o come, a leas in some espec s, om he same popula ion as he
lea ning da ase (in p obabili y- heo y ja gon, pa ial o condi ional
exchangeabili y applies; see § 3.2). This p ecludes using he p esen
Bayesian-nonpa ame ic so wa e o o ecas how he s a is ics o he
ull popula ion could change in he u u e. Howe e , he so wa e can
65 Dy land e al. 2022b.
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