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Author(s): PA Thwaites, JQ Smith and RG Cowell
Article Title: Propagation using Chain Event Graphs
Year of publication: 2008
Link to published article:
http://www2.warwick.ac.uk/fac/sci/statistics/crism/research/2008/paper
08-06
Peter A.Thwaites JimQ. Smith RobertG. Cowell
StatistisDept. StatistisDept CassBusinessShool
Universityof Warwik UniversityofWarwik CityUniversity
CoventryUKCV47AL CoventryUKCV47AL London EC1Y8TZ
Abstrat
AChainEventGraph(CEG)isagraphialmodelwhihisdesignedtoembodyonditional
indepen-deniesinproblemswhose statespaesarehighly asymmetrianddonotadmitanaturalprodut
struture. Inthispaperwepresentaprobabilitypropagationalgorithmwhih usesthetopologyof
theCEGtobuildatransporterCEG.Intriguingly,thetransporterCEGisdiretlyanalogoustothe
triangulatedBayesianNetwork(BN)inthemoreonventionaljuntion treepropagationalgorithms
usedwithBNs. Thepropagationmethodusesfatorizationformulaealsoanalogousto(butdierent
from) theones using potentialson liquesand separatorsof theBN. It appears that themethods
will be typially more eÆient than the BN algorithms when applied to ontexts where there is
signiantasymmetrypresent.
1 INTRODUCTION
Basedonaneventtree,aChainEventGraph(CEG)isamoreexpressivealternativetoadisreteBayesian
Network(BN),embodyingolletionsofonditionalindependenestatementsin itstopology. InSmithand
Anderson (2008) it is shown not only how asymmetries in a problem's sample spae an be represented
expliitly through the topology of its CEG, but also how it an express a muh wider range of types of
onditional independene statementnot simultaneouslyexpressiblethroughasingle BN.As with the BN,
theCEGofanhypothesisedmodelanbeinterrogatedusingnaturallanguagebeforethegraphisembellished
with probabilities. In ThwaitesandSmith (2006)andRiomagno andSmith (2005)wedemonstratehow
the CEGanalso be used to representand analyse various ausal hypotheses. In this paperwe ontinue
thedevelopment ofCEGsby demonstratinghow thegraphprovidesauseful struturefor fastprobability
propagationinasymmetri models.
IthasbeennotedthattheCEGisanespeiallypowerfulframeworkforinferenewhenaprobabilitymodel
is highly asymmetri and eliited througha desription of how situations unfold. Although theoretially
a Bayesian Network an be used in this ontext, the lique probability tables are then very sparse and
ontainmanyzerosorrepeatedprobabilities. This impedesfastpropagation algorithmsandhasledto the
development of many ontext spei variantsof BNs (Boutilier et al 1996, MAllester et al 2004, Poole
andZhang2003,Salmeronetal2000),oftenbasedontreeswithinliques. Thesedevelopmentsprovokethe
questionasto whether asingletree might be used for propagation insteadof theBN. Now obviously the
eventtree itself expresses noonditional independenies in its topologyand these independenies are the
buildingbloksofurrentpropagationalgorithms. However,unliketheeventtree,theCEGexpressesafairly
omprehensiveolletionofonditionalindependenies. Inthispaperwedemonstratethesurprisingfatthat
thereisadiretanaloguebetweenadistributiononaBNexpressedasaprodutofpotentialssupportedbya
graphofliquesandseparators,andpropagationalgorithmsonCEGsusingthedistributionsonthehildren
oftheCEG'snon-leafnodesandmarginallikelihoodsonthevertiesthemselves. Thisenablesustodevelop
fast propagation algorithms that use a single graph, the transporter CEG { analogous to a triangulated
BN { as its framework. This framework is highly eÆient for asymmetri/non-produt-spae ontexts,
and inpartiular doesnotinvolvepropagating zerosin sparsebut largeprobabilitytables, norontinually
repeating thesamealulations,whih wouldbetheaseif wewereto usetheBN asaframework inthis
sortofenvironmentwithanaiveBNpropagationalgorithm.
probabilitytables assoiated with the hildren of agivenvertex of theCEG takethe role of liques, and
vertexprobabiliestaketherole of separators. Insetion 4wedemonstrate theeÆieny of thisalgorithm
withasimpleexample.
2 PROBABILITY TREES AND
CHAIN EVENT GRAPHS
Probability trees (Shafer 1996), and their ontrol analoguesdeision trees, havebeen found to be a very
natural and expressive framework for probability and deision problems, and they provide an exellent
frameworkfor desribingsample spae asymmetryandinhomogeneity in agiven ontext (seefor example
FrenhandInsua(2000)). WestartwithaneventtreeT withvertexsetV(T)and(direted)edgesetE(T).
Heneforthall thetree'snon-leafvertiesfvgsituations,anddenote thissetofvertiesS(T)V(T). We
anonvert anevent treeinto aprobability tree byspeifying atransition matrixfrom its vertiesV(T),
wheretheabsorbingstatesorrespondtotheleafverties. Transitionprobabilitiesfromasituationarezero
exeptfortransitionstooneofthatsituation'shildren. Thismakesthetransitionmatrixuppertriangular.
Suhamatrixwould lookliketheonein Table1whihshowspartofthematrixfortheproblemdesribed
inExample 1. Notethateahtransitionprobabilityanbeidentiedbyanedgeonthetree.
Table1: PartofthetransitionmatrixforExample1
v 0
v 1
v 2
v 3
v 1 4
v 2 4
v 3 4
v 1 5
v 2 5
v
1 1
v 0
0
1
2
3
0 0 0 0 0 0
v 1
0 0 0 0
5
0 0 0 0
4
v 2
0 0 0 0 0
6
0
7
0 0
v 3
0 0 0 0 0 0
8
0
9
0
. . .
. . .
. . .
Onewayofseeingonditionalindependene statementsonaBNisasidentitiesin ertainvetorsof
ondi-tionalprobabilties{expliitlythoseprobabilityvetorsassoiatedwithdierentanestorongurationsbut
thesameparentongurationofavariableintheBN(RiomagnoandSmith2007). There isalargelass
ofmodelswhere theprobabilitiesinsomeofthe rowsof thetransitionmatrixanbeidentitifed witheah
other. The CEGis a topologial representation of this lass of models, and the transporter CEG dened
belowisasubgraphoftheCEG.
LetT(v i
), i=1;2betheuniquesubtrees whose rootsarethe situationsv i
,and whih ontainallverties
afterv i
inT. Sayv 1
andv 2
areinthesamepositionwif:
1. thetreesT(v 1
)andT(v 2
)aretopologiallyidential.
2. thereis amapbetweenT(v 1
)andT(v 2
)suh that theedgesin T(v 2
)are annotated,underthat map,
bythesame(possiblyunknown)probabilitiesastheorrespondingedgesin T(v 1
).
ItiseasilyhekedthatthesetW(T)ofpositionswpartitionsS(T). Furthermore,somewhatmoresubtlely,
ifv 1
;v 2
2w andv
ij
2V(T(v i
)),thenthevertexsetsof T(v i
)i=1;2aremappedontoeahotherbythis
map, and v ij
2w
j
i= 1;2forsome position w j
(providing v ij
is notaleaf-vertexin either subtree). For
detailsofthis propertyseeSmithandAnderson(2008).
We now draw a new graph to depit both the sample spae of T and ertain onditional independene
statements. The transporter CEG C(T) is a direted graph whose verties W(C(T)) are W(T)[fw 1
g.
There is an edge (e 2 E(C(T))) from w 1
to w 2
6= w 1
for eah situation v 2
2 w
2
whih is a hild of a
xed representativev 1
2w
1
forsomev 1
2S(T),and anedgefrom w 1
to w 1
foreah leafnode v2V(T)
whihisahildofsomexedrepresentativev 1
2w
1
forsomev 1
2S(T). ThetransporterCEG(heneforth
labelledsimplyas C)isatually thesubgraphofaCEG(denedin SmithandAnderson(2008))where all
undiretededgesin theCEGare omitted. TherelationshipbetweenthetransporterCEGandthe CEGis
diretlyanalogousto therelationshipbetweenatriangulatedBNandtheoriginal BN.Certainonditional
independenestatementsthatanbelostthroughonditioningaresimplyforgottensothatanhomogeneous
propagation algorithmanbe onstrutedonthebasisof theenduringonditional independenies. Unlike
the BN, this CEG an have many edges between two verties and always has a single sink vertex w
1 .
unfoldings ofthehistoryof aunit) arein oneto oneorrespondene with theset ofroot to sinkpaths on
thetransporterCEG.TheCEG-onstrutionproessisillustratedinExample1.
Example1
Considerthetreein Figure1,whihhas16atoms(root-to-leafpaths). Notethatasthesubtreesrootedin
thevertiesfv i 4
gare thesame,and thoserooted in fv i 5
gare thesame,thedistribution on thetreeanbe
storedusing7onditional tableswhih ontain16(9 free)probabilities.
Our transporter CEG (Figure 2) is produed by ombining the verties fv
i 4
g into one position w 4
, the
vertiesfv i 5
gintoonepositionw 5
,thevertiesfv i 6
ginto oneposition w 6
, andallleaf-vertiesintoasingle
sink-nodew 1
.
θ
1
=
0
.6
θ
10
θ
11
v
0
v
1
v
2
v
3
v
4
1
v
5
1
v
4
2
v
5
2
θ
10
= 0.
65
θ
11
= 0.25
θ
2
= 0.25
θ
12
= 0.1
θ
11
θ
3
= 0
.1
5
θ
10
θ
14
= 0.7
v
4
3
θ
6
= 0.7
θ
5
= 0.35
θ
12
θ
13
= 0.3
θ
4
= 0.65
θ
7
= 0.3
θ
8
= 0.4
v
inf
1
v
inf
2
θ
9
= 0.6
θ
12
θ
13
θ
14
θ
15
= 0.65
θ
16
= 0.35
θ
15
θ
16
v
6
1
v
6
2
Figure1: TreeforExample1
w
0
w
2
w
1
w
inf
w
3
(
θ
3
)
θ
2
= 0.25
θ
1
=
0
.6
θ
8
=
0.
4
θ
7
= 0
.3
θ
4= 0.65
θ
3
=
0
.1
5
θ
11
= 0.2
5
θ
5
= 0.35
θ
6
= 0
.7
(
θ
1
θ
5
+
θ
2
θ
6
+
θ
3
θ
8
)
w
4
w
5
(
θ
2
θ
7
+
θ
3
θ
9
)
θ
12
= 0.1
θ
9= 0.6
θ
10
= 0.65
θ
13
= 0
.3
θ
14= 0.7
(
θ
1
)
(
θ
2
)
w
6
(
θ
2
θ
7
θ
14
+
θ
3
θ
9
θ
14
)
θ
15
=
0
.6
5
θ
1
6
=
0
.3
5
Figure2: TransporterCEGforExample1
Thefull CEGforourexample(as dened in Smithand Anderson(2008))issimple {ithasno undireted
edges, and is idential to the transporter CEG C. Fora simple CEG, all the onditional independenies
inherentin theproblemareonveyedbythetransporterCEG.
Figure 2 shows the probabilities of reahing eah position w (the event reahing w, denoted (w), is
the union of all root-to-sink paths passing through w). It also shows eah edge-probability e
(w 0
j w)
(= ((e(w;w
0
)) j (w)), where (e(w;w
0
)) is the union of all root-to-sink paths utilising the
edgee(w;w 0
)).
Theonditional independeniesembodied in thisgraphannot beeÆientlyodedin aBNwithout
intro-duingtables withmanyzeros. Soevenin thisverysimpleexamplewehaveeÆienygains instoringthis
ALGORITHM
3.1 THE FRAMEWORK
Tospeifythejointdistributionofallrandomvariablesmeasurablewith respettoaCEGwesimplyneed
to speify the vetorof onditional probability mass funtions assoiated with eah of its positions. The
rst step of our propagation algorithm is analogous to the triangulation step for a BN, whih allows us
to retain all onditional independene properties at the ostof apossiblelossof eÆieny. Todo this we
ignoreonditionalindependene statementsodedbytheundiretededgesoftheCEGandwork onlywith
thesubgraphonsistingofitspositions,togetherwithitsdiretededges,but notitsundiretededges{our
transporterCEGC.
Foreahposition w2W =W(C)nfw 1
gwestoreavetorofprobabilites(w)=f e
(w 0
jw)je(w;w 0
)2
E(w)gwhereE(w)E(C)isthesetofalledgesemanatingfromw. (w)isofourseaonditional
proba-bilitydistribution. WeletX(w)betherandomvariabletakingvaluesonf1;2;:::;n(E(w))g(wheren(E(w))
isthe numberofedges emanatingfrom w)whose probabilitymassfuntion is givenbythe omponents of
(w) taken in order. The positions w 2 W takethe role of the liquesin a triangulated BN, whilst the
vetorsf(w)jw2Wgareanalogoustotheliqueprobabilitytables.
We an now speify the probability
of every atom (a root to sink path of C, of length n()) as a
funtionof f(w)j w2WgandC. If:
=(w
0
=w
[0℄;e
[1℄;w
[1℄;:::;e
[n()℄;w 1
)
then
= n()
Y
i=1 (e
[i℄)
where (e
[i℄) is a omponent of the probability vetor (w
[i 1℄), 1 i n(). It follows that the
distributionofanyrandomvariable measurablewithrespettoC anbealulatedfromf(w)jw2Wg.
3.2 COMPATIBLE OBSERVATIONS
ReallthatpropagationalgorithmsforBNsbasedontriangulationareonlydesignedtopropagateinformation
that anbeexpressedin theform O(A) =fX j
2A
j
gforsomesubsetsfA j
gofthesample spaesoffX j
g
the vertex-variables of the BN. Propagating information about the value of some general funtion of the
vertexvariables using loal messagepassing is not generally possible, beause onditioning on the values
of suh afuntion an destroy theonditional independenies onwhih theloal stepsof the propagation
algorithmdependfortheirvalidity.
Inthesamewaythetypesof observation weaneÆientlypropagate usingC andf(w)jw2Wgneeds
to beonstrained. Ingeneralanobservationanbeidentied withasubsetof thesetof allroot tosink
pathsfg. Themost obviousonstraining assumptionon (andthe onewe willheneforth makein this
paper) about what we might learn is that our observation an be identied with having learned that
fX(w)2 A(w)g forsome subsets fA(w)g of the sample spaes of theposition random variables fX(w)g.
CallsuhasetC ompatible. NotethatisC ompatibleifandonlyifthereexistspossiblyemptysubsets
fE
(w)jw2Wgsuhthat
=fje
2E
(w)forsomew2W; for eahedge
e
onthepathinCg
SoweanidentifyaompatibleobservationwiththesetofedgesE
= S
w2W E
(w)E(C). Wenotethat
thesetof ompatibleobservations islargeand inpartiularwhen theCEGisexpressibleasaBNontains
allsetsoftheformO(A) denedabove.
Example2
Consider:
=fje
2fe
1 (w
0 ;w
1 );e
2 (w
0 ;w
2 );e
4 (w
1 ;w
1 );
e 5
(w 1
;w 4
);e 6
(w 2
;w 4
);e 7
(w 2
;w 5
);e 10
(w 4
;w 1
);
e 11
(w 4
;w 1
);e 14
(w 5
;w 6
);e 15
(w 6
;w 1
w
0
w
2
w
1
w
inf
w
4
w
5
w
6
Figure3: Subgraph forevent inExample2
3.3 MESSAGE PASSINGFROM
COMPATIBLE OBSERVATIONS ON
ACEG
The message passing algorithm is a funtion from the original probabilities f(w) j w 2 Wg to revised
probabilitieson thesamegraph f^(w)j w 2Wgonditional on theobservation. Note that one
edge-probabilities have been revised, the resulting graph may not be a minimal CEG (in that we may have
vertieswithin the graphwhih are theroots of idential sub-graphs). Weanadd anal algorithm step
(seebelow)toprodueaminimalCEGifthisisrequired.
Messagesare passedfrom theterminal edgesbakwardsthrough thetransporterCEGalong neighbouring
edges until reahing the root in a ollet step givinga new pair f(w);(w) j w 2 Wg. We then move
forwardfromtherootproduingrevised f^
(w)j w2Wg.
LetW( 1) denotethesetofpositionsallof whoseoutgoingedgesterminatein w 1
inC.
1. Foranyedge e(w;w 1
)suh thatw2W( 1), setthepotential e
(w 1
jw)=0ife(w;w 1
)2=E
, and
e
(w 1
jw)=
e (w
1
jw)ife(w;w 1
)2E
. Lettheemphasis:
(w)=
X
e2E(w)
e (w
1 jw)
Saythat w 1
andeahof thesepositionsisaommodated.
2. For any position w all of whose hildren are aommodated, and edge e(w;w
0
), set the potential
e
(w 0
j w) = 0 if e(w;w 0
) 2= E
, and e
(w 0
j w) =
e (w
0
j w) (w
0
) if e(w;w 0
) 2 E
. Let the
em-phasis:
(w)=
X
e2E(w)
e (w
0 jw)
Saythat wisaommodated.
3. Repeatstep2until allw2W areaommodated.
Thisompletestheolletsteps.
4. Forallw2W,set:
^
(w)=0 if(w)=0
^
(w)=
(w)
(w)
if(w)6=0
where(w)=f
e (w
0
jw)je(w;w 0
)2E(w)g.
Clearlywehavethat:
^ e
(w 0
jw)=0 ife(w;w 0
)2=E
^ e
(w 0
jw)=
e (w
0 j w)
(w)
ife(w;w 0
0
theform:
^
(w jw 0
)= Y
i=0 ^ e
(w i+1
jw i
)
andweget:
^
((w))=
X
2f(w 0
;w)g ^
(wj w 0
)
AspreviouslynotedthegraphC
soproduedisnotneessarilyminimal. Itispossible(althoughunneessary
for information-propagatingpurposes)to add a further stepto the algorithm to make theadjustmentsto
produeaminimal CEG.Thissteprequires that anyvertiesthat arenowequivalentareombinedintoa
singleposition.
Fromthedenition ofaommodation,theorderofthese operations(liketheperfetorder usedtoupdate
atriangulatedBN)depends onlyonthetoplogyofC,so itanbeset upbeforehand.
Example3
Steps1,2and3giveusthegraphinFigure4:
0.579 + 0.1916
w
2
w
1
w
inf
0.3
x
0
.4
55
0.65
0.25
0.35
x 0.9
0.7
x 0.
9
0.65 + 0.25
w
4
w
5
0.455
0.65
0
.6
5
0.65 + 0.315
0.63 + 0.1365
w
0
0.
6
x
0.
96
5
0.25 x 0.7665
w
inf
w
6
0.65
0.7 x 0.65
Figure4: Potentialsandemphasesadded
Step4givesustheCEGinFigure5(notethatourCEGisagainsimple,andalsominimalwithouttheneed
fortheadditionalsteppreviouslymentioned).
w
2
w
1
w
inf
0.65 / 0.965 = 0.674
0.315 / 0.965 = 0.326
w
4
w
5
0.65 / 0.9 = 0.722
1
w
0
0.579 / 0.771 = 0.751
0.192 / 0.771 = 0.249
0.25 / 0.9 = 0.278
0.63 / 0.767 = 0.822
0.137 / 0.767 = 0.178
w
6
1
Figure5: UpdatedCEGC
0 1
( j)=^()=
n() Y
i=1 ^ (e
[i℄)=
n() Q
i=1 (e
[i℄)
n() 1
Q
i=0 (w
[i℄)
Alsonotethat at theostofsomeomputation,weanperforminfereneonthereduedgraphC
whose
edgesE(C
)arejust theedges ein E(C) withnon-zeroprobabilities(e),^ and whosevertiesW(C
) are
thew2 W(C) forwhih (w)6=0. Thenon-zero edgeand vertex probabilitiesof C then simplymap on
to their orresponding edge andvertex probabilitiesin C
. Note that, unlike for theBN, any non trivial
C ompatible observationstritlyredues thenumberofedgesintheedgesetafter thisoperation.
Softwareimplementationofthealgorithm isalreadyunder way{apseudo-odeversionisgivenbelow:
LetC(W(C);E(C)) beatransporterCEG withedges in E(C) having labelse i
; i =1;2;:::n e
, suh that
i<j )e i
e
j (e
i
does notliedownstreamof e j
onanyw 0
! w
1
path); and positions in W(C)nfw 1
g
having labelsw i
; i =0;1;2;:::m w
, suh that i < j ) w i
w
j
. Toupdate the edge-probabilities on C
followingobservationofanevent,do:
(1)SetA=
(2)SetB=
(3)Seti=1
(4)Repeat
(a)Selete i
(b)Ife i
2E
,adde
i toA
otherwise,set ^ e
i =0
()Seti=i+1
Untili=n e
+1
(5)Set(w
1 )=1
(6)Setj=m w
(7)Repeat
(a)Seletw j
(b)Repeat
(i)Selet e(w j
;w 0 j
)2E(w
j )\A
(ii)Set e
(w 0 j
jw j
)=
e (w
0 j
jw j
)(w
0 j )
(iii)Adde(w j
;w 0 j
)toB
UntilE(w
j
)\AB
()Set(w
j )=
P
e2E(w j
)
e (w
0 j
jw j
)
(d)Setj =j 1
Untilj = 1
(8)Foreah e(w;w 0
)2E
,set ^ e
(w 0
jw)=
e (w
0 jw)
(w)
(9)Returnf^ e
g
4 A CLOSER LOOK AT OUR
EXAMPLE
Considerthe CEG in Figure 2 and let the16 edges be labelled e i
in the sameorder as thef i
gthereon.
ThisCEGrepresentsaTreatmentregimeforaseriousmedialondition,andtheedgesarrythemeanings
givenin Table2:
Table2: Edgedesriptors
Edge Desription
e 1
Not ritial{TreatmentpresribedI
e 2
Liverfailure {Treatment::: II
e 3
Liver&Kidneyfailure{Treatment:::II
e 4
5
e 6
;e 8
Respondsto II {Surgery:::III
e 7
;e 9
NoresponsetoII{Surgery:::IV
e 10
Reovery{Lifetimemonitoring
e 11
Reovery{Lifetimemediation
e 12
;e 13
Deathinsurgery
e 14
SurvivessurgeryIV{Treatment:::V
e 15
Reovery{LifetimeontreatmentV
e 16
NoresponsetoV{Dies
ItisnotpossibletorepresentthisregimeeÆientlyasaBN,noryetasaontext-speiBN,giventhatthe
asymmetryoftheproblemdoesnotjustlieinithavingasymmetrisamplespaestrutures. Byequatingthe
desriptionsofedgese 4
ande 10
;edgese 11
ande 15
;andedgese 12
;e 13
ande 16
,weanhoweverapproximate
theproblemwitha4-variableBN;whereX 1
Diagnosisandinitial treatmentantakevaluesorresponding
totheoutomesfNot ritial, Liverfailure, Liver &Kidney failureg;X 2
2nd treatmentto fNone,III,IVg;
X 3
3rdtreatmentto fNone, Vg;andX 4
Responseto fDeath,Partial reovery, Fullreoveryg. The BNfor
thisapproximationtothemodelisgivenin Figure6.
X
1
X
3
X
2
X
4
Figure6: BNforourexample
TostorethemodelusingaCEGrequires16ells(orrespondingtothe16edges),butinthisBN27ells(9
fortheliquefX 1
;X 2
gand18forfX 2
;X 3
;X 4
g),14ofwhiharestoringthevaluezero.
TheeventinourexampleorrespondstotheobservationthatapatientwasnotdiagnosedwithLiverand
Kidney failure, and is still alive. Propagation of this event enablesa pratitionerto establishprobability
distributionsforthepossiblehistoriesofourpatient. Notethatitisonlythefatthatweandesribein
suhasimplemannerthathasallowedustoapproximatetheproblem withtheBNin Figure6.
PropagatingoftheeventusingasimpleJuntionTreealgorithmontheliquesoftheBNtakesaminimum
of43operations. PropagationontheCEGusingouralgorithmrequires32operations(orrespondingto 16
bakwardedges,6bakwardvertiesand 10forwardedges). Soeveninthissimpleexample,usingtheCEG
ismoreeÆientthantheBN.TheeÆienyhereisduemainlytothefatthattheliqueprobabilitytables
ontainmanyzeros. ThisisreetedintheCEGbythew 0
!w
1
pathsnotallhavingthesamelength. It
isthisformofasymmetryinamodelthatontext-speiBNsdonotopewithadequately,andwhyCEGs
areabetterstrutureforusewiththistypeof problem.
The problems in whih the algorithm desribed above are most eÆient are when the CEG struture is
known tobesimple. Tostoretheprobabilitytables forthe CEGrequiresonly N =#(W(C))+#(E(C))
<2#(E(C))ells. InthisasetheolletstepinvolvesonlyN alulationsandthetopologyoftheCEGis
validsothat in partiulartheoriginalprobabilitytable strutureanbepreserved. Thepotentialprodut
neessitatesonlyasingledistribute stepwhihagain onlyinvolvesat mostN alulations. Forlargetrees
withmuhofthetypeofsubtreesymmetrydisussedabovethepropagationisextremelyfast.
AnalogouslytotheexamplequotedbySmithinthedisussionofLauritzenandSpiegelhalter(1988),onsider
this very simple example arising from model seletion in graphial orpartition model problems, an area
urrently attrating some interest: Consider a model with random variables X 1
;:::X n
, where X
1 with
M =
1 =
2
(n 1)(n 2) possible states, determines whih pair of binary variables from fX
2 ;:::X
n g are
dependent,allothervariablesfromfX 2
;:::X n
gbeingindependentofeahotherandofthepairdetermined.
TheCEGofthismodelhasatmostM(1+2n)edgesand2+Mnpositions,whereastheBNisasinglelique
requiringM2
n 1
ellsforstorage. AsthenumberofoperationsrequiredforpropagationonboththeBN
andtheCEGis ofthesameorderofmagnitudeasthenumberofellsrequiredfor storage,itislearthat
ThereareseveraladvantagesofthismethodovertheodingofthistypeofproblemthroughaBN.Firstly,
the alulated probabilities anbe projeted bak on to the edges of theeliited tree, so that the
onse-quenesofinferenesgivendierenttypesofinformationanbeimmediatelyappreiatedbythepratitioner.
Seondly,theaommmodationofdataintheform ofaompatibleobservationis muhmoregeneralthan
the aommodationof subsetsof observationsfrom a predeterminedset ofrandom variables, sothe CEG
providesamoreexibleframeworkforpropagation,partiularlywhendataisontingentlyensored. Thirdly,
thereare eÆienygainsasoutlinedabove. Weintendtoshowhowgreatthese gainsanbeforverylarge
problemsin alaterpaper.
Notealsothat,asistheasewiththetriangulationstepinBN-basedalgorithms,therearefasteralgorithms
(Thwaites2008)thantheonedesribedabove,althoughtheylosesomeofthisalgorithm'sgenerality.
OfourseBNsprovideasimplerrepresentationofmoresymmetriproblemsandshouldalwaysbepreferred
when thethree ontingeniesare not satisied. TheCEG doesnot provide auniversal improvement over
theBNforpropagation. Inpartiular in problemswhentheunderlying BNisdeomposablebut theCEG
isnotsimpletheBNpropagationanbemuhmoreeÆient. Butinhighlyasymmetriproblems,theCEG
shoulddenitelybearsthoie.
ItshouldbenotedthatitisalsopossibletodeneadynamianalogueoftheCEG,andourinvestigationof
thesesuggeststhatatime-sliedCEG(analogoustoatime-sliedBN)willbeanidealvehileforadynami
updatingalgorithm. Wehopeto reportonthesedevelopmentsinthenear-future.
ThealgorithmdesribedaboveisurrentlybeingodedbyCowellwithinfreelyavailablesoftware,andwill
beavailableshortly.
APPENDIX
Welaimthat:
^ e
(w 0
jw) , ((e(w;w
0
))j ;(w))
= (
e
(w 0
jw)
(w)
ife(w;w 0
)2E
0 ife(w;w
0 )62E
Proof:
ForaCEGC,andC ompatibleevent,letT bethetreeassoiatedwithC,T
thetreeassoiatedwith
C
,andT ()
thesubtreeofT ontainingonlythoseroot-to-leafpathsin. T ()
diersfromT
inthatthe
formerretainstheedge-probabilitiesfrom T.
Considerapositionw2C (w 2C
)orrespondingtoasetof vertiesfv i
g2T. Thenthesubtrees rooted
ineahv i
areidentialbothintopologyand intheiredge-probabilities.
Ifthereisasubpath(w 0
;w)whihisnotpartofaw 0
!w
1
pathin(ie. (w
0
;w)existsin C,but not
inC
)thentherewillexistasubsetoffv i
gwhihdoesnotexistinT
(orT ()
). Wesplitthesetfv i
ginto:
fv i
g i2I
vertiesexistingin T
fv i
g i2J
vertiesnotexisting inT
Beause is C ompatible, the subtrees in T
()
rooted in eah v i
2 fv i
g i2I
are also idential both in
topologyand intheiredge-probabilitiesthat theyretain fromT.
Suppose there exists an edge e(w;w 0
) in C, then for eah v i
2 fv i
g, there exists an edge e(v i
;v 0 i
) in T
orrespondingtothisedge. Notethat:
(w)=
[
i2I[J (v
i )
(e(w;w
0 ))=
[
i2I[J (e(v
i ;v
0 i
))
e
(v 0 i
j v i
)=
e (w
0
j w) 8i2I[J
andsinethesubtrees inT ()
rootedin eahv i
2fv i
g i2I
areidential,wealsohave:
(j(v
i
))=(j(v
(;(e(v i
;v i
))j(v i
))=(;(e(v
j ;v
j ))j(v
j ))
fori;j2I
[( j (v
i
)) is the sum of the probabilities of all the (v
i ;v
leaf
) subpaths in T
() , and (;(e(v i ;v 0 i
))j(v i
))isthesumoftheprobabilitiesof allthe(v i ;e(v i ;v 0 i );v 0 i ;v leaf
)subpathsinT () ℄ So: ^ e (w 0
jw)=((e(w;w
0
))j;(w))
=
(;(w);(e(w;w
0 ))) (;(w)) = (; S i2I[J [(v i );(e(v i ;v 0 i ))℄) (; S i2I[J (v i ))
(anexpressionevaluatedonT)
sine(v
i
)\(e(v j
;v 0 j
))=fori6=j
= P i2I[J (;(v i );(e(v i ;v 0 i ))) P i2I[J (;(v i ))
But\(v
i
)= forv i
2fv i
g i2J
,sothisequals: P i2I (;(v i );(e(v i ;v 0 i ))) P i2I (;(v i )) = P i2I (;(e(v i ;v 0 i
))j (v i ))((v i )) P i2I
(j(v
i ))((v i )) = (;(e(v j ;v 0 j
))j(v j )) P i2I ((v i ))
(j(v
j )) P i2I ((v i ))
foranyv j 2fv i g i2I = (;(e(v j ;v 0 j
))j(v j
))
(j(v
j ))
foranyv j
2fv i
g i2I
Turningourattentionto thetermsin thealgorithm,welaimthat (w)=(j (v i
))and
e (w
0 jw)=
(;(e(v
i ;v
0 i
))j (v i ))(v i 2fv i g i2I
)forall w;e(w;w 0
)2C
, where fv i
g i2I
isthe setof vertiesin T ()
orrespondingtow. Weprovethisbyindution:
Step1.
Considerpositionsw2W( 1). Then:
(w)= X e e (w 1 jw)=
X e e (w 1 jw) = X e e (v leaf jv i
) in T
()
foranyv i
2fv i
g i2I
=( j(v
i ))
Step2.
Suppose w is suh that all of its outgoing edges terminate in positions fw
0
g for whih
(w 0
)=(j(v
0 i )). Then: (w)= X e e (w 0 jw)=
X e e (w 0
jw)(w
0 ) = X e e (v 0 i jv i
)(j(v
0 i
))
foranyv i
2fv i
=
e
((e(v i
;v 0 i
))j (v i
))(j(v
0 i
))
But(v
0 i
)=(e(v
i ;v
0 i
))(v
i
)inatree, sothis
equals:
X
e
((e(v i
;v 0 i
));(v 0 i
)j(v i
))
(j(v
i );(e(v
i ;v
0 i
));(v 0 i
))
= X
e
(;(e(v
i ;v
0 i
));(v 0 i
)j(v i
))
= X
e
(;(e(v
i ;v
0 i
))j(v i
))
=(;(v
i )j (v
i
))=(j(v
i ))
Hene:
e
(w 0
jw)=
e (w
0
jw)(w
0 )
=
e (v
0 i
jv i
)(j(v
0 i
))
foranyv i
2fv i
g i2I
=((e(v
i ;v
0 i
))j(v i
))( j(v
0 i
))
=:::=(;(e(v i
;v 0 i
))j(v i
))
Wenowombineourtworesultstogive:
^ e
(w 0
jw)=
(;(e(v
j ;v
0 j
))j(v j
))
(j(v
j ))
=
e (w
0 j w)
(w)
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Intel-ligene,172:42{68,2008.
[2℄ C.Boutilier, N. Friedman, M.Goldszmidt, and D. Koller. Context-spei independene in Bayesian
Networks.InProeedingsofthe12thConfereneonUnertaintyinArtiialIntelligene,pages115{123,
Portland,Oregon,1996.
[3℄ R.G.CowellandA. P.Dawid. Fast retrationofevidene inaprobabilistiexpert system. Statistis
andComputing,2:37{40,1992.
[4℄ S.FrenhandD. R.Insua. Statistial DeisionTheory. Arnold, 2000.
[5℄ S.L.LauritzenandD. J.Spiegelhalter. Loalomputations withprobabilitiesongraphialstrutures
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[9℄ E.M.RiomagnoandJ.Q.Smith.Thegeometryofausalprobabilitytreesthatarealgebraially
on-strained.InL.PronzatoandA.Zhigljavsky,editors,OptimalDesignandRelatedAreasinOptimization
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