PERSONALIZED PROGNOSIS & TREATMENT
USING LUSTED-JAYNES MACHINES:
AN EXAMPLE STUDY ON CONVERSION
FROM MILD COGNITIVE IMPAIRMENT TO ALZHEIMER’S DISEASE
P.G.L. Po a Mana
wes e n no way uni e si y o applied sciences, no way
I. Rye
uni e si y o oslo, no way
A. Vik
haukeland uni e si y hospi al, no way
M. Kociński , A. Lunde old , A.J. Lunde old
uni e si y o be gen, no way
A.S. Lunde old
wes e n no way uni e si y o applied sciences, no way
9 h Feb ua y 2023
Co espondence should be sen o
E-Mail: pgl@po amana.o g
Psychome ika Submission Feb ua y 9, 2023 2
PERSONALIZED PROGNOSIS & TREATMENT USING LUSTED-JAYNES MACHINES:
AN EXAMPLE STUDY ON CONVERSION
FROM MILD COGNITIVE IMPAIRMENT TO ALZHEIMER’S DISEASE
Abs ac
The p esen wo k p esen s a s a is ically sound, igo ous, and model- ee algo i hm –
called “Lus ed-Jaynes machine” in homage o hese wo pionee s – o use in pe sonalized
medicine. The algo i hm is designed i s o lea n om a da ase o clinical 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 he algo i hm 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 algo i hm 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.
Key wo ds: Clinical decision making, U ili y heo y, P obabili y heo y, A i icial In elli-
gence, Machine Lea ning, Base- a e allacy
Psychome ika Submission Feb ua y 9, 2023 3
1. In oduc ion: Pe sonalized p ognosis, ea men , and compu e algo i hms
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 a compu e algo i hm.
Besides a sha ed diagnosis o Mild Cogni i e Impai men and associa 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 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
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”.
Psychome ika Submission Feb ua y 9, 2023 4
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 compu e algo i hms: pe sonalized inpu and ou pu
In he example abo e, we said “in hese asks, he clinician will be helped by a compu e
algo i hm”. 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
an assis i e compu e algo i hm be designed in o de o ake ully in o accoun pa ien di e ences?
Al hough he example abo e conce ns speci 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 auxilia y o supple-
men 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 op imal ea men . An
assis i e algo i hm 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
Psychome ika Submission Feb ua y 9, 2023 5
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 algo i hm ha mee s he equi emen s
abo e. This algo i hm 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
3
, 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
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:
(a) he clinical-p edic o alues o he cu en pa ien ;
(b)
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;
(c) 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:
(a)
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 4a;
(b) inal p ognos ic p obabili ies, gi en inpu s 4a–4b;
(c) op imal ea men , gi en inpu s 4a–4c;
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 4a–
4b only, ge a p ognos ic p obabili y 5b as ou pu , and hen p oceed o ea men assessmen by
o he means o algo i hms.
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
3
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 .
Psychome ika Submission Feb ua y 9, 2023 6
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.
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 heo 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
4
. Wi hou ea u e 3, he capabili y o auxilia y con ex ual
in o ma ion, he algo i hm 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 8is 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 2and 10, he ac ha his algo i hm 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 algo i hm 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.
We call his algo i hm a Lus ed-Jaynes machine, o easons explained in he nex sec ion. I is
a he momen a ailable as a collec ion o sc ip s
5
in he R p og amming language (R Co e Team,
2023), which we plan o assemble in o a clinician- iendly R package soon.
The me hodology unde lying his algo i hm has been success ully demons a ed o Alzheime ’s
Disease wi h a smalle numbe o p edic o s (An oniano-Villalobos e al.,2014), is used in many
applica ions in as ophysics (E en Ho izon Telescope Collabo a ion,2019,2022;Del Pozzo e al.,
2018), and i s ad an ages in neu oc i ical ca e ha e ecen ly been emphasized (Jawa and Maslo e,
4see e.g. Edmonds e al. (2015,2020), whose me hods we ind, howe e , inconclusi e.
5doi:10.17605/os .io/zb26 ,h ps://gi hub.com/pglpm/ledley-jaynes_machine.
Psychome ika Submission Feb ua y 9, 2023 7
2023).
The nex sec ion 2gi es an in ui i e unde s anding o he Lus ed-Jaynes machine’s unde lying
p inciples and wo kings. The machine’s conc e e applica ion 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 e
6
, which also
explains how o use he R sc ip s.
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. The Lus ed-Jaynes machine
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 machine’s applica ion. I
can be ead a e § 3by eade s who would like o see he machine 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 obabili 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, auxilia 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.7
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, inciden ally, died o Alzheime ’s Disease (Shah e al.,2013) –
and Lus ed (Ledley and Lus ed,1959a,b,1960;Lus ed and Ledley,1960;Lus ed,1967), who also
p omo ed i s algo i hmic implemen a ion (Lus ed,1968;Ledley,1959,1960, § 1-5 p. 21). Clinical
decision-making is oday explained and exempli ied in b illian ex books o medical s uden s and
clinicians (Weins ein and Finebe g,1980;Sox e al.,2013;Hunink e al.,2014). An ou line is gi en
in § 3.3.
6h ps://gi hub.com/pglpm/ledley-jaynes_machine/ aw/main/omni-p edic o _machine.pd
7
Jaynes (2003, chs 13–14); on Neumann and Mo gens e n (1955); Cox (1946); Sa age (1972); Luce and Rai a
(1957); Rai a and Schlai e (2000); Rai a (1970); Lindley (1988); K eps (1988).
Psychome ika Submission Feb ua y 9, 2023 8
The “Lus ed-Jaynes machine” is an algo i hmic implemen a ion, as d eamed by Lus ed and
Ledley
8
, 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 . The name is a homage o Lus ed and o Jaynes (2003), who b illian ly explained
he induc i e logic unde lying such a “ obo ”.
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 (Russell and No ig 2022, pa IV; Jaynes
2003, chs 1–2, 13–14). The Lus ed-Jaynes machine 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, he Lus ed-Jaynes machine 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.
9
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 obabili y dis ibu ion, he Lus ed-Jaynes machine
can indeed calcula e any quan i y ou pu ed by o he machine-lea ning algo i hms. Fo example ( o
e minology see e.g. Mu phy,2012, § 8.6):
•
“Disc imina i e” algo i hms: he p obabili y p(
Y|X
)o any se o p edic ands
Y
gi 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
X
gi en any se
o p edic and alues Y.
Mo e gene ally, he machine 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
8c . he Appendices in Lus ed 1968.
9
This goes by he Sibylline echnical name o “Bayesian nonpa ame ic densi y eg ession”; see e.g. Rod íguez e al.
(2009); Bha acha ya and Dunson (2010); and Walke ’s (2010) wi y o e iew.
Psychome ika Submission Feb ua y 9, 2023 9
unc ion
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−
(
X
)]. Thus he Lus ed-Jaynes machine 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, he machine 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 he Lus ed-Jaynes machine 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 models, he Lus ed-Jaynes machine 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 ionally 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, andom o es s, suppo - ec o machines, logis ic eg ession, o gene alized
linea models, he Lus ed-Jaynes machine 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 o he hypo hesis space will be a ound
he ex emum (MacKay,1992a;Mu phy,2012, ch. 16; see also Sel and Cheeseman,1987). The
unde lying necessi y o a mo e ex ensi e su ey, howe e , becomes mani es in many o he obliga o y
p ocedu es ha go oge he wi h he aining o mos machine-lea ning algo i hms, c oss- alida ion
being a p ominen example (MacKay,1992b). This leads o a hi d special cha ac e is ic o he
Lus ed-Jaynes machine: i does no need alida ion se s, es se s, o o he da a spli s; no does i
need c oss- alida ion p ocedu es. In ui i ely his is he case because he unde lying hypo hesis-space
su ey ealizes a so o ull- ledged c oss- alida ion and da a pa i ion. I can indeed be p o en
ha one o he in e nal compu a ions o he machine is ma hema ically equi alen o doing
k
- old
c oss- alida ions o all possible da a spli s and k(Po a Mana,2019;Fong and Holmes,2020).
Such lexibili y and in o ma ionally ich ou pu come, o cou se, a a compu a ional cos . Un il
some yea s ago, he cos would ha e been p ohibi i e in all bu he simples in e en ial p oblems. Bu
oday an in e ence p oblem in ol ing 13 a ia es and 700 da apoin s, such as he example conside ed
in he p esen wo k, akes less han six hou s o compu a ion on an o ice compu e wo ks a ion. We
discuss compu a ional limi a ions u he in § 4.2.
Psychome ika Submission Feb ua y 9, 2023 16
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 he Lus ed-Jaynes machine 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 he Lus ed-Jaynes machine.
Gi en hese p edic o alues he Lus ed-Jaynes machine can ou pu any p obabili ies o in e es o
he clinician. 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:15
•
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
16
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.
The Lus ed-Jaynes machine 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 Hippocampal 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
answe 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 (e.g. Be na do and Smi h,2000, §§ 4.2–4.3), p o ided he pa ien and
da ase can be conside ed as belonging o he same popula ion.
15
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%.
16p(A|B)is he p obabili y o Agi en B, as well as he likelihood o Bgi en A(Good,1950, § 6.1).
Psychome ika Submission Feb ua y 9, 2023 17
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.
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 Hip-
pocampal 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 e-
quency o con e sion o ad in he subpopula 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
).
Psychome ika Submission Feb ua y 9, 2023 18
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 compu ed by gene alizing associa ions be ween p edic o s and p edic and 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 .
Psychome ika Submission Feb ua y 9, 2023 19
Le us explain wi h a amilia example why pa icula associa ions
canno be gene alized. Mos s uden s o medicine lea n abou he base- a e
allacy (Ba -Hillel,1980;Jenny e al.,2018;Sp enge and Weinbe ge ,
2021;Ma hews,1996). Conside a la ge se o clinical ials, illus a ed in
he uppe able on he side, whe e each do ep esen s, say, 10 000 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), 71.4% o hem (o 5/7, uppe
squa e) e en ually de eloped a disease. The allacy lies in judging ha a
new eal pa ien om he ull popula ion, who 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 33.3% (5/15), 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
p edic o alue “−”.
The e is a 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, 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 samples o he ials (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 (P o os ,2000;D ummond and Hol e,2005;
Weiss and P o os ,2003). Mo e gene ally his disc epancy can appea whene e a popula ion and
a sample da ase om i 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 equencies may be elied upon.
Conside he ull popula ion. Among all pa ien s who de eloped he disease, 83.3% o hem (o 5/6,
uppe ow) had he p edic o alue “+”, while among hose who did no de elop he disease, 33.3%
(o 1/3, lowe ow) had he 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 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
Psychome ika Submission Feb ua y 9, 2023 20
by D, and he ull-popula ion base a e by B. Bayes’s heo em yields
p(Y|X, D, B) = p(X|Y, D)·p(Y|B)
P
Y
p(X|Y, D)·p(Y|B)(2)
In ou example we ind
p(Y= + |X= +, D, B) = p(X= + |Y= +, D)·p(Y= + |B)
p(X= + |Y= +, D)·p(Y= + |B)p(X= + |Y=−, D)·p(Y=− | B)
≈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 he p obabili y p(
p edic and |p edic o s,da ase
) om he da ase 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 app op ia e, and allow us o apply co ec ions when use
o he la e is inapp 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. Mu phy,2012, § 8.6) 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 (Lindley and No ick 1981;Quin ana e al. 2017;Sox e al. 2013, ch. 4; Hunink e al.
2014, ch. 5). 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 ognoses. I is su icien ha some condi ional s a is ics may be
applicable o he speci ic pa ien . Fo a pa ien coming om a di e en egion, o example, i may be
Psychome ika Submission Feb ua y 9, 2023 21
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 obabili 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 coin oss always p edic con e sion p edic o s |p edic and & base a e
37.3% 50% 63.3% 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
(“gene 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 he Lus ed-Jaynes machine, a e incapable o ex ac ing his use ul in o ma ion.
Psychome ika Submission Feb ua y 9, 2023 22
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. See Lindley (2014,
especially a ound §§ 7.3, 8.6; 1981) o a clea and logically impeccable p esen a ion no obscu ed
by echnical language (mo e echnical e e ences a e de Fine i 1930,1937;Dawid 2013;Be na do
and Smi h 2000, §§ 4.2–4.3, 4.6; see also Malinas and Bigelow 2016,Sp enge and Weinbe ge 2021
abou con ounding and Simpson’s pa adox, o which his opic is igh ly ela ed).
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,
he Lus ed-Jaynes machine 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 Yand 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.
Psychome ika Submission Feb ua y 9, 2023 23
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
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 be ad isable o unde go he ea men e en i he disease is less p obably p esen
han no . I he clinician adminis e he ea men o 100 simila pa ien s, and 20 o hem de elop
he disease, hen he clinician has added 5
×
20 = 100 and sub ac ed
1
12 ×
60 = 5 yea s o hei
combined li espans.
In bo h examples, i is clea ly impo an o assess he p obabili y – ha ing p ecise connec ions
wi h he popula ion equency – ha he pa ien will de elop he disease. In he i s example, he
Psychome ika Submission Feb ua y 9, 2023 24
ea men should only be adminis e ed i he p obabili y is highe han 83.3%; in he second, i can
be adminis e ed i he p obabili y is a leas 1.6%. The Lus ed-Jaynes machine, as explained in he
p e ious sec ions, ells he clinician he speci ic p obabili y o he cu en pa ien .
Bu he choice be ween ea men s depends no only on he p obabili y o he medical condi ion.
He e is whe e di e ences be ween pa ien s a y and ma e he mos . Conside again he second
example abo e, abou he less dange ous ea men . Le us add ha he ea men would ex end he
li espan by i e yea s, bu would also somewha wo sen he quali y o li e o he pa ien and o he
pa ien ’s amily. Suppose ou pa ien is qui e old and i ed, has had a happy li e, and is now looking
wi h a peace ul mind owa ds dea h as a na u al pa o li e. Such a pa ien may p e e o o ego he
bo he o he ea men and he addi ional i e yea s, e en i he p obabili y o he disease is qui e
high.
The bene i s o he di e en ea men s, and he p obabili y h esholds a which one ea men
becomes p e e able o ano he , mus he e o e be judged and quan i ied p ima ily by he pa ien .
U ili y heo y and maximiza ion o expec ed u ili y allow clinician and pa ien o make such
judgemen s and decisions in a cohe en way (Sox e al. 2013;Hunink e al. 2014;Lus ed 1968; see
also he clea and cha ming exposi ion by Lindley 1988, and O’Hagan e al. 2006).
We summa ize he main, pa ien -dependen p ocedu e o decision-making, and show how ou
compu a ions so a i pe ec ly wi h i .
The clinician i s assesses and lis s he mu ually exclusi e cou ses o ac ion a ailable o he
speci ic pa ien . These could be p e en i e o cu a i e ea men s, mo e es s, doing no hing, and so
on. O en he e a e sequences o decisions a ailable, bu he u ili y amewo k can be applied o hem
as well (see e e ences abo e and Rai a,1970). In he p esen wo k we a e calling hese he e ogeneous
al e na i es simply “ ea men s” o simplici y (see oo no e 2, p. 3). The lis ea men s is al eady
pa ien -dependen : some al e na i es may no be medically sui able (say, owing o alle gies o o he
clinical condi ions), some may be economically oo cos ly, and so on.
Each ea men will ha e di e en consequences, which addi ionally depend on he pa ien ’s
unknown clinical condi ion o in e es . A ea men may ha e some consequences i he pa ien
has o will de elop he disease, and di e en consequences o he wise. The pa ien quan i ies, wi h
he clinician’s guidance, he bene i s and cos s – echnically called “u ili ies” – o such possible
consequences. The quan i ica ion o u ili ies is no wi hin he scope o he p esen wo k. The
e e ences ci ed abo e o e guidelines and ules o nume ically ansla ing ac o s such as quali y o
li e and expec ed li espan in o u ili ies.
The ea men s, unce ain clinical condi ions, and he quan i ied u ili ies
U
o hei consequences
can be o ganized in o a able o his o m:
Psychome ika Submission Feb ua y 9, 2023 25
clinical condi ion aclinical condi ion b. . .
ea men αUαa Uαb . . .
ea men βUβa Uβb . . .
. . . . . . . . . . . .
which can be compac ly ep esen ed by a so-called u ili y ma ix
Uij
), he ow index
i
enume a ing
he ea men s, and he column index
j
he clinical condi ions. No e ha he numbe o possible
ea men s and clinical condi ions do no need o be equal; gene ally, hey a e no .
The expec ed u ili y
¯
Ui
o a ea men
i
is calcula ed as he expec a ion o i s u ili ies
Uia, Uib, . . .
wi h espec o he p obabili ies p(a),p(b), . . . o he clinical condi ions a, b, . . . :
¯
Ui:=Uia p(a) + Uib p(b) + · · · (4)
No e ha his co esponds o a ma ix mul iplica ion be ween he ma ix o u ili ies and he ec o
o p obabili ies.
Finally, he ecommended ea men is he one ha ing maximal expec ed u ili y.
Applica ion o he example s udy
A p esen he e a e no cu es o Alzheime ’s Disease, al hough some ecen pha macological
agen s a e shown o delay onse o pa hology ela ed o Alzheime ’s Disease
17
. Bu o he sake o
ou case s udy le us imagine ha in he nea u u e he e a e h ee mu ually exclusi e ea men
op ions o p e en ion o e a da ion o he disease; call hem
β
,
γ
,
δ
, he simple op ion o “no
ea men ” being deno ed by
α
. The clinical condi ions o be conside ed a e jus wo: he pa ien
will con e o Alzheime ’s Disease (
cAD =Y
), o will emain wi h s able Mild Cogni i e Impai men
(cAD =N).
We ha e he e o e 4
×
2=8possible consequences o he ou ea men s depending on he wo
clinical condi ions. Ou ou pa ien s and he clinician quan i y he u ili ies, a i ing a he u ili y
ma ices shown in able 5, op. Oli ia, A iel, and Cu is quan i y he bene i s o he ea men s in
exac ly he same way, bu Bianca’s quan i ica ion di e s sligh ly, because o he in e ac ion o he
ea men s wi h se e al alle gies and addi ional clinical condi ions, as explained in § 1.0.
The p obabili ies o he wo medical condi ions a e hose ound in he p e ious subsec ion, able 4.
Fo b e i y, we deno e jus by p(
cAD
) he p obabili y o con e sion gi en a pa ien ’s p edic o alues,
17
e.g. lecanemab, a monoclonal an ibody in usion gi en e e y wo weeks, a ge ing amyloid be a plaques;
see
h ps://www. da.go /news-e en s/p ess-announcemen s/ da-g an s-accele a ed-app o al-alzheime s-
disease- ea men .
Psychome ika Submission Feb ua y 9, 2023 32
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).
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 n-
ing da ase . Compa e his igu e wi h ig. 3, p. 15,
bo om- igh .
Psychome ika Submission Feb ua y 9, 2023 33
The Lus ed-Jaynes machine 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. (2022) in o pe spec i e. The 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 m-
a 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
(Weins ein and Finebe g,1980;Sox e al.,2013;Hunink e al.,2014) a e he pionee ing wo k o
Lesley & Lus ed (1959a;1959b;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 connec 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. 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 common 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
Psychome ika Submission Feb ua y 9, 2023 34
Table 6: Summa y o he clinician’s pa ien -dependen inpu s and he Lus ed-Jaynes machine’s
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 F F F M
HV/10−34.26 4.26 4.26 [missing]
APOE4 NNNY
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
Addi ional in o ma ion
auxilia y in o none amily his o y, base a e none none
applicable da ase 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
Ou pu s o Lus ed-Jaynes machine
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
Psychome ika Submission Feb ua y 9, 2023 35
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 algo i hm, he “Lus ed-Jaynes machine”, ha mee s all hese equi e-
men s (§ 2) and ca ies ou he calcula ions equi ed by decision heo y. This algo i hm 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 ecommended
o he s udy o Alzheime ’s Disease (Temp e al.,2021; see also asa,2016,2019), and ha e been
success ully demons a ed in a simple p edic o se ing (An oniano-Villalobos e al.,2014). 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 k (Rye e al.,2022).
The Lus ed-Jaynes machine was also shown o ha e uses ha go beyond 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 a
Lus ed-Jaynes machine 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 Lus ed-Jaynes
machine 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, because he machine 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 he Lus ed-Jaynes machine
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.’s (2013) and Hunink e al.’s (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.
Psychome ika Submission Feb ua y 9, 2023 36
This ques ion is answe ed by a ma hema ical heo em a he e y co e o decision heo y
20
,
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 obabili 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 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’ ?
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.’s (2013) and Hunink e al.’s (2014).
4.2. Range o applica ion o Lus ed-Jaynes machines
Fi s le us emphasize, e en i i is ob ious, ha he quali y o he esul s ob ained wi h he
Lus ed-Jaynes machine 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 he Lus ed-Jaynes machine 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 he Lus ed-Jaynes machine has wo kinds o bounds: compu a ional
and heo e ical.
The ac ha he Lus ed-Jaynes machine 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
20
Sa age (1972); Luce and Rai a (1957); Rai a and Schlai e (2000); A kinson e al. (1964); Fe guson (1967); Lindley
(1988,1977); K eps (1988); Be na do and Smi h (2000); P a e al. (1996); Lindley (2014); Pe ig ew (2019)
Psychome ika Submission Feb ua y 9, 2023 37
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 a Lus ed-Jaynes machine, as a pos -p ocesso o hei aw ou pu . The machine ex ac s use ul
in o ma ion usually hidden in hei ou pu a a low compu a ional cos (Dy land e al.,2022b); 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 he Lus ed-Jaynes machine’s 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 Lus ed-Jaynes machine o o ecas how
he s a is ics o he ull popula ion could change in he u u e. Howe e , he machine can be used
o ime-dependen (longi udinal) in e ences wi hin a s able popula ion, such as o ecas s o he
u u e ime o disease onse , expec ed li eleng h, and simila . Fo ins ance, i da a abou he ime o
con e sion o Alzheime ’s Disease we e a ailable in he da ase , he Lus ed-Jaynes machine could
o ecas no only whe he , bu also when he con e sion could ake place (c . e.g. De la C uz-Mesía
e al.,2007).
Finally, he machine is no mean o handle sequences o decisions in a clinical decision ee (Sox
e al.,2013, ch. 6; Hunink e al.,2014, ch. 1) – i would be impossible in a pe sonalized app oach,
because such a ee is ully pa ien -dependen – bu i could be used in indi idual decision b anches.
A. Fu he ma hema ical and compu a ional de ails abou he example applica ion o
he Lus ed-Jaynes machine
The Lus ed-Jaynes machine su eys he space o possible dis ibu ions o equencies o all 13
a ia es discussed in § 3.0, o he ull popula ion o pa ien s om which he da ase o igina es (see
§3.2). In he p esen s udy, i does so by ma hema ically ep esen ing a gene ic join equency
dis ibu ion F(Y, X1, X2, . . . )as a con ex mix u e o app op ia e ke nels p oduc s:
F(Y, X1, X2, . . . ) = X
i
wiK(Y|υi)K(X1|ξi
1)K(X2|ξi
2)· · · ,
along he ideas in Dunson and Bha acha ya (2011) and Ishwa an and Za epou (2002); see also
Rossi (2014); Rasmussen (1999). This ep esen a ion uses a o al o 1535 independen pa ame e s
(
wi,υi,ξi
1, . . .
), wi h oughly 190 pa ame e s o each con inuous o in ege a ia e. As a c ude
in ui ion, i is as i we di ided he ange o each a ia e in o 190 bins, and conside ed all possible
equency his og ams o e hese. The ac ual pa ame iza ion is sma e , inc emen ally using
Psychome ika Submission Feb ua y 9, 2023 38
pa ame e s o ep esen less and less smoo h ai s o he dis ibu ion. We indeed expec he
dis ibu ion o a ull popula ion o ha e some deg ee o smoo hness, owing o physical and biological
easons. Ac ually, he numbe o pa ame e s used is in p inciple in ini e, because he machine gi es
a wa ning i he da a indica e ha mo e pa ame e s a e needed. In he p esen s udy, he da a
indica e, on he con a y, ha ewe han 250 pa ame e s would be enough. No e ha he machine
cons uc s he ke nels
K
(
|
)and hei p oduc au oma ically, depending on how many and wha
kinds o a ia es he da ase comp ises.
The p obabili y o a candida e equency dis ibu ion
F
is de e mined by i s “ i ”
F
(
D
)o he
da a Dand a p io -expec a ion ac o p(F), as explained in § 2.2:
p(F|D)∝F(D) p(F).
Finally, he p edic i e condi ional p obabili y o any wo se s o a ia es
Z′, Z′′
, gi en he da ase , is
gi en by he expec a ion o e he unknown
F
, as equi ed by he p obabili y calculus and de Fine i’s
heo em (Be na do and Smi h,2000, § 4.6):
p(Z′|Z′′, D) = ZF(Z′|Z′′) p(F|D) dF
whe e he condi ional equencies F(Z′|Z′′):=F(Z′, Z′′)/F(Z′′).
The equency-space su ey and he calcula ion o he p obabili ies p(
F|D
) o he popula ion-
equency dis ibu ions was done ia Gibbs sampling (Neal,1993, ch. 4; MacKay,2005, § 29.5; Casella
and Geo ge,1992) wi h he R package Nimble (de Valpine e al.,2021), using 1024 independen
Ma ko chains. S a iona i y was assessed by common diagnos ic measu es (Gilks e al.,1998),
especially in eg a ed au oco ela ion ime (Ch is en and Fox,2010) and Hellinge dis ance (Boone
e al.,2014), as well as isual inspec ion. An au oma ed me hod o s a iona i y check was de eloped,
o be discussed in u u e publica ions.
Con lic o In e es S a emen
The au ho s decla e ha he esea ch was conduc ed in he absence o any comme cial o inancial
ela ionships ha could be cons ued as a po en ial con lic o in e es .
Au ho Con ibu ions
The au ho s we e oo imme sed in he de elopmen o he p esen wo k o keep a de ailed eco d
o who did wha .
Psychome ika Submission Feb ua y 9, 2023 39
Funding
The s udy was suppo ed by g an s om he T ond Mohn Resea ch Founda ion, g an numbe
BFS2018TMT0, and om The Resea ch Council o No way, p ojec numbe 294594.
Acknowledgemen s
PGLPM hanks Soledad Gonzalo Cogno and I án Da ido ich o inspi ing discussions; Maja,
Ma i, Mi i, Emma o con inuous encou agemen and a ec ion; Bus e Kea on and Sai ama o illing
li e wi h awe and inspi a ion; and he de elope s and main aine s o Nimble, L
A
T
E
X, Emacs, AUCT
E
X,
Open Science F amewo k, R, Inkscape, Lib eO ice, Sci-Hub o making a ee and impa ial scien i ic
exchange possible.
Compu a ions unde lying he Lus ed-Jaynes machine we e ini ially pe o med on esou ces
p o ided by Sigma2 – he Na ional In as uc u e o High Pe o mance Compu ing and Da a
S o age in No way (p ojec NN8050K).
Clinical-da a collec ion and sha ing o his p ojec was unded by he Alzheime ’s Disease
Neu oimaging Ini ia i e (ADNI) (Na ional Ins i u es o Heal h G an U01 AG024904) and DOD
ADNI (Depa men o De ense awa d numbe W81XWH-12-2-0012). ADNI is unded by he
Na ional Ins i u e on Aging, he Na ional Ins i u e o Biomedical Imaging and Bioenginee ing, and
h ough gene ous con ibu ions om he ollowing: AbbVie, Alzheime ’s Associa ion; Alzheime ’s
D ug Disco e y Founda ion; A aclon Bio ech; BioClinica, Inc.; Biogen; B is ol-Mye s Squibb
Company; Ce eSpi , Inc.; Cogs a e; Eisai Inc.; Elan Pha maceu icals, Inc.; Eli Lilly and Company;
Eu oImmun; F. Ho mann-La Roche L d and i s a ilia ed company Genen ech, Inc.; Fuji ebio; GE
Heal hca e; IXICO L d.; Janssen Alzheime Immuno he apy Resea ch De elopmen , LLC.; Johnson
& Johnson Pha maceu ical Resea ch De elopmen LLC.; Lumosi y; Lundbeck; Me ck Co., Inc.; Meso
Scale Diagnos ics, LLC.; Neu oRx Resea ch; Neu o ack Technologies; No a is Pha maceu icals
Co po a ion; P ize Inc.; Pi amal Imaging; Se ie ; Takeda Pha maceu ical Company; and T ansi ion
The apeu ics. The Canadian Ins i u es o Heal h Resea ch is p o iding unds o suppo ADNI
clinical si es in Canada. P i a e sec o con ibu ions a e acili a ed by he Founda ion o he
Na ional Ins i u es o Heal h (www. nih.o g). The g an ee o ganiza ion is he No he n Cali o nia
Ins i u e o Resea ch and Educa ion, and he s udy is coo dina ed by he Alzheime ’s The apeu ic
Resea ch Ins i u e a he Uni e si y o Sou he n Cali o nia. ADNI da a a e dissemina ed by he
Labo a o y o Neu o Imaging a he Uni e si y o Sou he n Cali o nia.
So wa e-a ailabili y S a emen
The R sc ip s used o his s udy can be ound in he Open Science F amewo k p ojec doi:
10.17605/os .io/zb26
(see also he eposi o y
h ps://gi hub.com/pglpm/ledley-jaynes_mac
Psychome ika Submission Feb ua y 9, 2023 40
hine). We hope o assemble hem in o an R package soon.
Re e ences
Al a ez-Melis, D. and B ode ick, T. (2015). A ansla ion o “The cha ac e is ic unc ion o a andom
phenomenon” by B uno de Fine i. a Xi doi:
10.48550/a Xi .1512.01229
. T ansl. o de Fine i
(1929)
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