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Author(s): JQ Smith, E Riccomagno and PA Thwaites
Article Title: Causal analysis with Chain Event Graphs
Year of publication: 2009
Link to published article:
http://www2.warwick.ac.uk/fac/sci/statistics/crism/research/2009/paper
09-08
JimQ. Smith a
Eva Riomagno
b
Peter Thwaites
a
a
DepartmentofStatistis,UniversityofWarwik,Coventry,CV47AL,
UnitedKingdom
b
DepartmentofMathematis,UniversitadegliStudidi Genova,
ViaDodeaneso35,16146Genova,Italy
Abstrat
AstheChainEventGraph(CEG)hasatopologywhihrepresentssetsof
onditional independenestatements,it beomes espeially usefulwhen
problemslienaturallyinadisreteasymmetrinon-produtspaedomain,
or whenmuhontext-speiinformation is present. Inthis paperwe
showthatitanalsobeapowerfulrepresentationaltoolforawidevariety
ofausalhypothesesinsuhdomains. Furthermore,wedemonstratethat,
aswithCausalBayesianNetworks(CBNs),theidentiabilityoftheeets
ofausalmanipulations whenobservationsofthe systemare inomplete
anbeveriedsimplybyreferenetothetopologyoftheCEG.Welose
thepaperwithaproofofaBakDoorTheoremforCEGs,analogous to
Pearl'sBakDoorTheoremforCBNs.
Keywords
BakDoortheorem,BayesianNetwork,ausalmanipulation,ChainEvent
Graph,onditionalindependene,eventtree,graphialmodel.
1 Causal Manipulation
Muhreentworkin theeld ofausalityhasfoussedonhowauserelatesto
ontrol,andtheanalysisofontrolledmodels. Here,with theadvoatesofthis
approah we assume the existeneof abakground idle systemwhih is then
subjetedto somesortofinterventionormanipulation.
The Bayesian Network (BN) has been one of the most suessful graphial
toolsforrepresentingomplexdependenyrelationships,andunsurprisingly
re-searhershavelookedtointerpretthediretionalityoftheedgesoftheBNasin
somewayausal. ThishasledtothedevelopmentoftheCausalBayesian
Net-work (CBN), usinganon-parametri representationbasedonstrutural
equa-tionmodels[11℄[17℄[18℄[26℄. Theseprovideaframeworkforexpressingassertions
latedandsomeofitsvariablesareassignedertainvalues.
Wearguethat for manyausalproblems, theBNis notthemost appropriate
graphialmodel. Therearetwomainreasonsforthislaim:
Firstly, manyproessesannot be satisfatorilydesribed bya BN.Examples
ouringenomis,epidemiologyandmulti-agentsystems;furtherexamplesan
be found in [2℄[15℄[20℄. Suh proesses tend not to admit a natural produt
spaestruture{theyareasymmetriinthesensethatmeasurementvariables
may have dierent olletions of possible outomes given dierent vetors of
valuesfor setsofanestralvariables,leadingifone usesaBN, tosparselique
probability tables with manyzeros orrepeatedprobabilities. Many problems
mayhavesomevariableswhihhavenooutomesgivensomevetorsofvalues
ofanestralvariables.
Seondly, even forthose problemswhose idle settings ansatisfatorilybe
de-pitedasaBN,thereare manymanipulations whih annotbedesribed
ade-quatelyviathisrepresentation.
Theproblemswithwhihweareonernedeitherexhibitsigniantasymmetry
in the representation of theiridle state, orare subjet to non-symmetri
ma-nipulations.Whereproblemsdonotdisplaysuhasymmetries,theBNremains
theappropriatehoieforrepresentationandanalysis.
The BN provides a simple way of representing the dependene relationships
between the measurement variables of a problem, but annot express
graphi-allyalltheontext-speiorsamplespaeinformationneededforanaurate
representationofamoreasymmetri problem. Othermoreprimitive toolsare
neededhere,andwenotethatdespitetheproliferationofgraphialmodelsover
thelast twodeades,therststagein thedevelopmentofamodelisstilloften
basedontheeliitationofaneventtree. Althoughtopologiallyomplex,event
treeshaveseveraladvantagesoverBNs,inluding(i)theyexpliitlyaknowledge
asymmetriesembeddedinastruture,bothinitsdevelopmentandinitssample
spaestruture,(ii)theirsemantisaremuhlosertomanyverbaldesriptions
oftheworld,espeiallywhenthosedesriptionsrevolvearoundhowthings
hap-pen rather than how the world appears. These advantages are ompellingly
argued, in for example[23℄[18℄[26℄ in the ontext of ausality. In the related
eldofdeisionanalysis,FrenhandInsua[10℄arguethattheadvantagesof
in-uenediagramsoverdeisiontreesareillusory,andpointoutthatasymmetri
problemsin whihapartiularhoieof ationatadeisionnodemakes
avail-abledierenthoiesofationatsubsequentdeisionnodesthanthoseavailable
afteran alternativehoiearetheruleratherthantheexeption.
TheCBNissomewhatofahybridstruture,retainingarepresentationofsome
oftheonditional independene struture of theidle systemwhilstalso
repre-sentingmanipulationsasthesettingofertainmeasurementvariablestospei
values. Butitmaynotneessarilybetheasethattheinterventionsofinterest
orrespondtosettingtheoriginalvariablesintheidlesystemtospei values
notesthatinterventionsmayinvolvepoliieswherebyavariableX respondsto
someothervariable(s)Z througheitherafuntionalrelationshipx=g(z),ora
stohastirelationshipwherebyX is setto xwith someprobabilitydependent
onthevalue(s) z. Wean see that evenasymmetriidle systemangiverise
toahighlyasymmetristruturegivenapartiularmanipulationruleorpoliy.
Note alsothat notallvertiesin aBN aremanipulable in the sense that any
manipulation an be given a real interpretation. Although eets of a ause
anbereasonablyrepresentedbyarandomvariable, attimesthespeiation
ofaauseasthevalueofarandomvariableanbeartiial. Causesaremore
naturallyrepresentedasonditioningevents. Suhonditioningisnotelegantly
expressedin theBN,but issimply andintrinsiallydesribed inatree.
Anal-ogousarguments aremade byDawid [5℄ whoarguesthat auses are deisions
andnotdeisionrules.
There are partial solutions to someof these problems: Context-spei
vari-antsofBayesNetsexist[2℄[22℄[20℄[16℄, usuallywithtree-struturedonditional
probability tables annexedto thevertiesof aBNto allowfor theanalysis of
ontext-speiindependeneproperties. Thereisalsoanarttodrawingthe
ap-propriateBNofaproblemanditissometimespossibletoredenethevariables
enoding the problem or add more edges on the graph to aid representation.
Thismayprodueagraphonsistentwitha desriptionofaproess, but this
graphwillstillbeonlyapartialrepresentationingeneral. Theontext-spei
BayesNetissimilarlynotauniversalpanaea{anyproess(suhasatreatment
regime)whose unfolding depends on thestateof thesystemat anypartiular
pointandthevaluesofspeiovariatesatthatpoint,annotbeeÆiently
ex-pressedasaontext-speiBN,althoughitanalwaysbeexpressedeÆiently
asatree. Inpartiular,ontext-speiBNsdonotopeadequatelywiththose
problemswhere somevariableshavenooutomesgivensomevetorsof values
ofanestralvariables.
We have already noted that the event tree is auseful tool for representing
asymmetriproblems. Italsohasitsusesfortheausalanalysisofasymmetri
problems and the analysis of the eets of asymmetri ausal manipulations.
There is a lear link between the analysis of ontrolled models and the eld
of deision analysis. The ommon denition that A is a ause of B if the
probability of B given a manipulation to A is greater than the probability of
B given a manipulation to not A (see for example [18℄) learly suggests an
event-based(asopposedtovariable-based)approahtoausalanalysis;andan
obviousinitialandidateamonggraphialmodelsforsuhananalysismightbe
thedeisiontree. Weanthinkofaausalmanipulationasthemakingofsome
deision (possibly more than one), and as Frenh and Insua [10℄ note, suh
manipulationsoftenindue asymmetryinaproblem. Againthissuggeststhat
atreewould beasensiblerepresentation.
Byusing the framework of eventtrees the denition of manipulativeause is
from any diret link with the measurementproess. Using trees we an also
hoose the level of detail we inlude in our representation, and this an be
dependentonwhat weintendtodotothesystem. Weaninorporateontext
spei information that is informative about various ausal hypotheses (see
forexample [7℄). This ispartiularly useful in modelsof biologialregulatory
mehanisms,whihtypiallyontainmanynoisyandandorgates[24℄.
In[24℄ weintroduedanalternativegraphialmodel{theChainEvent Graph
(CEG), onstruted from anevent treetogether with a set of exhangeability
assumptions. It an beseenas ageneralisationof a probability graph[3℄[23℄,
and typially has many fewer nodes than the original event tree. The CEG
retainsthoseadvantagesthat eventtreeshaveoverBNsforthe representation
ofasymmetri problems;but theyarealso muh moreexible andusefulthan
eventtrees,sinetheirnodesrepresentintrinsieventsintheproblemandtheir
edgesdependeniesbetweenthem.
CEGs havetwo prinipal advantages over BNs for the representation of
(un-manipulated) asymmetri disrete problems. They express topologially all
theonditionalindependene strutureassoiatedwithaproblem {this isnot
bolted on aswith ontext-spei BNs. Theyalsoexpress samplespae
infor-mationgeneratedbytheasymmetryoftheproblem{againthisinformationis
expressedinthetopologyofthegraph. See[29℄foranexampleofaverysimple
problemwithanelegantrepresentationasaCEG,butwhihanonlybevery
lumsilyrepresentedbyaBN.
We present here a ausal extension to CEG models, whih we believe to be
as transparentand ompelling asthe extensionfrom BNsto CBNs. Setion 2
desribestheonstrutionofaCEGandontainsanexampleofhowan
asym-metri problem an be depited using suh a graph. We have not inluded
a formal denition of the CEG here; suh a denition an be found in [24℄,
wherealsoanbefoundmoredetailonreadingCEGsforonditional
indepen-deneproperties. Setion 3 introduesthe manipulation of these graphs, and
thistheoryisdevelopedinsetion4wherewelook atidentifying theeetsof
manipulations. Setion 5introduesaBak Doortheoremfor CEGs,a
gener-alisation of Pearl's Bak Door theorem for BNs [18℄. The idealised examples
throughoutthepaperanreadilybegeneralisedtomorerealistisenarios.
2 Chain Event Graphs
2.1 Derivation
TheCEGisafuntionofaneventtree[23℄,andinthissetionwedemonstrate
howtheCEGisderivedfromthistree.
An eventtree is adireted,rooted tree T, with vertex set V(T)and edge set
path-algebraofT),andlabelthedierentpossibleunfoldingsofthedesribed
proess. Events measureable with respet to this spae are unions of these
atoms.
Eah situation v serves as an index of a random variable X(v) whose values
desribethenextstage ofpossibledevelopmentsoftheunfoldingproess. The
statespaeX(v)of X(v)anbeidentied bothwiththe set ofdireted edges
e(v;v 0
)2E(T)emanating fromv in T and theset of end-nodes v 0
2V(T)of
theseedges. ForeahX(v)(v2S(T))welet
(v)=f(v
0 jv)jv
0
2X(v)g
and
(T)=f(v)g
v2S(T)
Afullspeiationoftheprobabilitymodelisgivenby(T;(T)).
Iftwosituationsvandv
2S(T)aresuhthattheirassoiatedrandomvariables
X(v)and X(v
) havethesamedistribution then wesay that v, v
arein the
samestageu{ifv;v
2u,andv 0
;v 0
labelthesameoutomegivenv;v
,then
(v 0
jv
)=(v
0
jv). ThesetofstagesL(T)formapartitionoftheset S(T).
Twosituationsv andv
arethereforein thesamestage whentheimmediate
futureevolutionfrombothvandv
isgovernedbythesameprobabilitylaw.
Inthe onversionof the event tree to the CEG, auseful interim graph is the
staged tree, dened formally in [24℄, whih is a oloured version of the event
tree: Ifastage u2L(T)ontainsasinglevertexv 2u,then edgesemanating
fromv arenotoloured,but ifuontainsmorethanonevertex,then alledges
emanatingfromeahv2uareoloured{twoedgese(v;v 0
);e(v
;v 0
)emanating
fromv;v
2uhavethesameolouriftheseedgeslabelthesameoutome(hene
(v 0
jv
)=(v
0 jv)).
Twosituations v and v
are said to bein thesame positionw if (i) alledges
onallsubpaths startingatv orv
areolouredin thestagedtreeofT,(ii)for
eah subpath in the set of subpaths emanating from v, the ordered sequene
ofolours is thesameas that fora subpath in theset of subpaths emanating
fromv
. Theset ofpositionsK(T)formsapartitionof theset S(T).
Two situations v and v
are therefore in the same position when the entire
futureevolutionfrombothvandv
isgovernedbythesameprobabilitylaw.
To eet the onversionof the staged treeinto aCEG, we start byhoosing,
for eah position w 2 K(T), a single representative situation v 2 S(T). For
eahedgee(v;v 0
)leavingvweonstrutasingleedgee(w;w 0
),wherew 0
=w
1
(a sink-node) if v 0
is a leaf vertex of T; otherwisew 0
is the position in K(T)
hosento representthesituationv 0
.
Theolouroftheedgee(w;w 0
)istheolouroftheedgee(v;v 0
)ifthisedgehasa
Positionsinthesamestagearethenonnetedbyundiretededges.
TheresultinggraphC(T)isalledaChainEventGraph{amixedgraphwith
vertexsetW(C)onsistingofthepositionsfrom K(T)andthesink-nodew 1
;
direted edge set E
d
(C) and undireted edge set E
u
(C) as desribed above.
Analogouslywith theeventtree,weallthesetofstagesoftheCEGL(C).
There isaone-to-one orrespondenebetween theroot-to-leafpaths in T and
theroot-to-sinkpathsinC(T). Eahatomof T beomesapath (w
0 ;w
1 )in
C(T),and thesepathsform theatomsofthe -algebraofthe CEG.Eventsin
C(T)areunions of w 0
!w
1
paths. Fortwopositionsw;w 0
2C(T) wewrite
ww
0
whenthere is adiretedpath in C(T)passingthroughw andw 0
, and
wpreedesw
0
onthispath.
Whenthesetof stagesL(T)of astagedtreeisidentialtothesetofpositions
K(T), weall C(T) simple. Simple CEGshavenoundireted edgesand sine
the olouring is therefore redundant, they an be treated as direted ayli
graphs. AnexampleofasimpleCEGanbefoundin [29℄.
EahstageuinourCEGC servesasanindexofarandomvariableX(u)whose
valuesdesribethenextstageofpossibledevelopmentsoftheunfoldingproess.
Thestate spaeX(u) of X(u)an be identied with theset of direted edges
e(w;w 0
)2E d
(C)emanatingfromanyw2u. ForeahX(u)welet
(u)=f(e(w;w
0
)jw)jw2ug
and
(C)=f(u)g
u2L(C)
Afullspeiationoftheprobabilitymodelisgivenby(C ;(C)).
2.2 Conditional independene
Theimplied onditional independene properties of astaged treean be read
from thetopologyofaCEG.These propertiesanappear asanumberof
dif-ferent typesof statement, and are dealt with in detailin [24℄ and [27℄. These
typesfallbroadlyintotwoategories{ut-basedproperties(developedin[24℄),
andposition-basedproperties (whih appear prinipallyin [27℄). Bynature of
itsevent-tree-basedonstrution,theremaybenointrinsisetofmeasurement
variables for the CEG over whih onditional independene is dened. This
allowsasigniantdegreeofexibilitytoouranalytialproedures.
We dene a olletion W of positions w 2 K(T) as a ne ut of C(T) if all
w 0
! w
1
paths in C(T) passthrough exatly one w 2 W; and we dene a
olletionU ofstagesu2L(T)asautofC(T)ifallw 0
!w
1
pathsinC(T)
passthroughexatlyonew2u2U.
The ut-based onditional independene properties of a CEG detailed in [24℄
topredithowtheproessisgoingtounfoldintheimmediatefuture. Seondly,
if we know that our proess has reahed some position w 2 W, then we do
notneedtoknowanythingabouthowitreahedwinordertopredithowthe
proessisgoingtobehaveduringitsompletefuture unfolding.
IfourCEGrepresentsasymmetrimodelwhihanbeperfetlydepitedbya
BN,thenweanprodueasequeneofutsandneutswhihgiveusexatly
thesamesetofonditionalindependenestatementsthatweoulddeduefrom
theBN[24℄. Inpratiehowever,in manyappliations(for exampleBayesian
deisionanalysis[9℄,riskanalysis[1℄,physis[14℄,biologialregulation[4℄)our
proesses are highly asymmetri, and the rst stage of model eliitation
pro-dues asymmetri event trees with root-to-leaf paths of unequal lengths and
eventspaes notadmittinganaturalprodut spaestruture. Insuh asesa
CEG-depitionof theproblem allowsfortherepresentation ofontext-spei
onditional independene statements that annot be shown on anunmodied
BN, and allows the analyst to dedue other ontext-spei onditional
inde-pendene properties that mightnot be apparentbefore theeliitation proess
isundertaken.
2.3 An Example
Thissetionontainsanexampleofamodelwiththetypeofasymmetri
stru-turedesribed above. Wedemonstrate how themodel anbe represented
a-urately using a Chain Event Graph, and disuss the diÆulties inherent in
representingthemodelviaaBayesianNetwork.
Example2.1 The polie hold a suspet S whom they believe threw a brik
through a shop window and stole a quantity of money. They wish to bring
S to ourt, but theremay be reasons for them notproeeding (suhas the lak
ofavailabilityofajudge; polie-forepoliyonthe amountofmoney needing to
bestolenbeforetheyarepreparedtopayfor forensitesting, ortakesuspetsto
ourt et). Whether they proeed or not an be thought of as outomes of an
indiator X
1
(with proeeding beinglabelledx 1 1
andnotproeeding labelledx 0 1 ).
It is unertain that the suspet was at the sene when the money was stolen
(indiator X
2
), that he was the individual who threw the brik and stole the
money (indiator X
3
), that the forensi servie will nd glass mathing the
windowglassonthelothingofS (indiator X 4
),thatawitnessW willidentify
S(indiatorX 5
),andwhetherSwillbeonvitedorreleased(theeetindiator
ofinterestX 6
).
ItwouldbeperfetlypossibletoonstrutoureventtreeandheneourCEGin
temporalordersothatedgesrepresentingtheoutomesofX 2
andX
3
preeded
thoseassoiatedwithX
1
,but if wesuppose that weare onstrutingourtree
througheliiting information frommembersof thepolie forethen X 1
is the
rst indiator of interest. In this our method is similar to that used in the
X 1
;X 2
;:::X 6
)oraausaltree(whereinthingshappeninatemporalorder). In
setion3 we look at theausal manipulation ofCEGs, where the topologyof
theCEGisalteredthroughadeisionofsomeomnisientdeisionmaker.
UnlessS isidentiedbythewitnessW,thenSwillnotbeonvited. Theglass
mathisbelievedonlytodependonwhetherSthrewthebrik;andthequality
ofthewitnessidentiationisbelievedtodependonlyonwhetherSwasatthe
sene of therime ornot. This is suÆient information for us to onstrut a
CEGfortheproblem. OurCEGisgivenin Figure1.
x
1
1
x
2
1
x
2
0
x
2
0
x
2
1
x
3
0
xxxx
x
5
1
x
5
0
x
5
0
x
5
1
w
10
x
4
0
x
4
1
x
5
0
x
5
0
x
3
1
x
4
0
x
4
1
x
3
1
x
3
0
x
5
1
x
5
1
w
7
w
9
w
8
w
0
w
1
w
2
w
3
w
4
w
6
w
5
w
11
w
12
w
13
w
14
w
inf
x
6
1
x
6
0
x
6
0
x
4
0
x
4
1
x
6
0
x
6
1
Figure1: CEGforExample2.1
As the reasonswhih might leadto thepolie not proeeding are notrelated
to their beliefs about S's presene at the rime sene et, we an see that
theprobabilitiesassoiatedwith edgeslabelledx 1 2
;x 0 2
;x 1 3
;x 0 3
are unaetedby
whether theysueededges labelled x
1 1
orx 0 1
. Hene the positions w
1
and w
2
in Figure 1are in the same stage (and so onneted by an undireted edge),
asare thepositionsw 3
andw
4
. Theposition w 3
representsthehistory(polie
proeed,S atsene). S ouldonlyhavethrownthebrikifhewasatthesene,
soedgeslabelledx 1 2
are sueededby edgeslabelledx
1 3
;x 0 3
, but edges labelled
x 0 2
arenot.
Ifthe polie donot proeed, then forensievidene is not olleted,and asS
isnottakento ourt,W will notbeaskedtotestify. Hene therearenoedges
labelledx 1 4
;x 0 4
;x 1 5
orx 0 5
onw 0
!w
1
pathsstartingwiththeedgex 0 1 .
ThesuessoftheforensitestbeingdependentonlyonwhetherornotSthrew
thebriktellsusthatthepositionsw 6
andw
7
onlyonwhetherS wasat therimeseneornottellsusthat thepositions w 8
and w
9
are in thesame stage, and that the positions w 10
and w
11
are in the
samestage.
If W does notidentify S (position w 13
), then the probability of onvition is
zero, and there is only oneedge e(w 13
;w 1
). If W does identify S, then the
probability of onvition depends on whether the forensi test wassuessful
(positionw 12
)ornot(positionw 14
). Thislastisnotexpliitinwhatthepolie
havetold us, but isapparentfrom the fat that thepolie would notpayfor
theforensitestifitwasnotgoingtobeanyuseto theminthease.
The detailing above of the possible developments of the ase amounts to a
desriptionoftheonditionalindependenestrutureoftheproblem,andlearly
mostoftheinformationprovidedisontext-spei. Figure1illustratesthefat
thatweareexpliitlyusingthetopologyof theCEGtoexpress theresulting
asymmetridependenystruture.
CouldwerepresentthisproblemusingaBN?Well,ofourseweould,but our
argument is that the CEG is a superior representation as it moreaurately
desribestheproblem, andis alsoamoresuitablegraphfor inferene,andfor
theanalysis ofausalmanipulation.
Ifweonsidertheproblemontingentonthepolieproeeding(ie. onditioned
onX 1
=1), weanprodueaBNonthevariablesX
2 ;X
3 ;:::X
6
whihis
on-sistentwith thepossibleunfoldings ofeventsdesribed above. Suh aBNan
onlybeapartialrepresentationasthesamplespaethatinludesX 1
isnot
nat-urallyaprodutspae. Thus(asalreadynoted)ifthepoliedonotproeedand
Sisreleased,forensievidenewillnotbeolleted,andthewitnesswillnotbe
allowedtotestify,sointhissensethesevariablesdonotexistunderthis
ontin-geny. ThisneednotstopustryingtodrawaBNoftheproblem,butweansee
thatsuh aBNwill notbeunique. Forexample, weouldmakethevariables
X 4
andX
5
tertiaryandlabeltheirextraoutomeswiththesymbol(tosignify
thattheonditionsforX i
takingvaluesorrespondingtox 1 i
orx 0 i
havenotbeen
met). We ouldarguethat one we knowthe valuesofX
4
and X
5
(inluding
X 4
;X 5
= 1
), we do not need to knowthe value of X 1
in order to make
as-sessmentsaboutX 6
. ThiswouldsuggestafullBNasinFigure2(a). Weould
alsoformallydenevaluesofX 4
;X 5
onditionedonX
1
=0,insuhawaythat
X 4
qX
1 j (X
2 ;X
3
) andX
5
qX
1 j (X
2 ;X
3 ;X
4
); whih mightleadusto aBN
asin Figure2(b).
Also, if wereturn to the CEG-representation of the problem in Figure 1, we
ouldinsist that everypathpasses throughanedge labelledwith outomesof
eah of X
1 ;X
2 ;:::X
6
by, whenever we need to add in an edge labelled with
outomesof X
3 ;X
4 ;X
5
,simplylabellingtheseedgeswithx 0 3
(S didnotthrow
thebrik {herebeausehe wasn'tat therime sene),x 0 4
(nomath isfound
byforensis{herebeausetheydidn't dothetest), x 0 5
(W didnotidentify S
{herebeausetheasedidnotgotoourt). Thiswouldalsogiveusaprodut
largeproportionofzeros(reetingtheatualasymmetryoftheproblem),with
the onsequene that our BN would be a very ineÆient way of storing the
informationdesribing theproblem. This would also meanthat anyattempts
topropagateinformationthroughthemodelwouldbeineÆientomparedwith
propagationmethodsavailablewithCEGs[29℄.
X
2
X
5
X
6
X
3
X
4
X
1
X
2
X
5
X
6
X
3
X
4
X
1
(a)
(b)
Figure 2: TwopossibleBNsforExample2.1
Morepertinentperhaps,isthatweanonlyeetthistransformationtoaBN
beauseourvertex-variablesaresimpleindiatorswithanoutometheeventof
interest does not happen. Many, ifnotmost, problemsin the areaspreviously
mentionedare moreomplex. Forexample, in aCEG representinga
disease-diagnostiproess,theoutomeslabellingtheedgesemanatingfromaposition
may be alistof thepossibleblood typesthat a patientmayhave, ora listof
theirpossibleombinationsofsymptoms. ToonvertsuhaproblemintoaBN,
additionaldummy outomes would need to be added to somevertex-variables
wheneverwehaveasetofroot-to-sinkpathsnotbeingallofthesamelength(for
example,ifapatientisrhesus +ve,wemaynothaveneededtoolletertain
informationaboutthepatient,astheirhanesofbeingaetedbysomedisease
is zero [28℄). These additions of dummy outomes in order to reate ourBN
resultin umbersomeandveryineÆientgraphialrepresentations.
A moredetailed disussion of the diÆulties involved in tting BNsto
asym-metriproblemsappearsin[29℄,whereinweshowthatevenforsmallproblems
ofthis type,the CEGismoreeÆientthan theBNasameansof storing the
model struture, but is also moreeÆient for the propagation of information
arossthesystem.
Buteven if wedo reate aBN-representation, it will still only onveyertain
aspetsofthestory. ThefatthatSanonlyhavethrownthebrik(X 3
=1)if
hewaspresentattherimesene(X 2
=1),orthefatthatonvition(X 6
=1)
requirespositivewitnessidentiation(X 5
=1)arenotexpressedintheBN.We
mightalsobeinterestedin theausaleetof,forexample,foringthewitness
toidentify S astheulprit(X 5
=1)ifamath intheglassisfound (X 4
=1).
Thisis notrepresentedin theusual semantis of ourBNsabove. Weouldof
ourse add an edge between the vertiesX
4
and X
5
in our BNs in Figure 2,
andthenthismanipulationouldbeexpressedasaontingentdeision,butwe
quatehere,butsuharepresentationwouldstillrequiretheadditionofdummy
outomes,andonditionalprobabilitytablesattahedtoverties{representing
informationthat isthere expliitely in thetopologyofourCEG. More
signi-antly, sineeahausalhypothesismayrequirethe additionofextraedgesto
ourBN, weannot representallpossiblehypothesesunder onsiderationwith
oneBN,withoutadisastrouslossofinformation{wemaywell needtoreate
newontext-speiBNsforeahdistintausalhypothesiswewishto
investi-gate. ThisisnotneessarywithourCEG.
3 Manipulating the Chain Event Graph
ACEGprovidesaexibleframeworkforexpressingwhatmighthappenwerea
modeltobemanipulatedormadesubjettosomeontrol.Suhamanipulation
resultsin amodiation(usually asimpliation)of thetopologyofour(idle)
CEGtoprodueamanipulatedCEG.Formanymanipulationsthismodiation
onsistssimplyof thepruning (removing)ofspeiededgesand positionsand
thereassignmentoftheprobabilitiesonasmall subsetofthediretededgesof
theCEG.
Disussions of ausal manipulation an be found in [12℄[18℄[23℄[26℄. Here we
followPearl [18℄ whosedo operatordesribesinterventionson direted ayli
graphs(DAGs):ThejointdensityfuntionofasetofrandomvariablesX 1
;:::X n
withsamplespaesX
1 ;:::X
n
fatorisesaordingtoaDAGas:
p(x 1
;:::x n
)= n Y
i=1 p(x
i j pa
i )
where p(x
i j pa
i
) is the probability of X i
taking the value x i
given that its
parents among X
1 ;:::X
n
takevaluesfrom x 1
;:::x n
. A random variable is
foredto assume a spei valuewith probability one, say X
j = x^
j
for some
j 2 f1;:::ng and x^ j
2 X
j
. A new density p( jj x^
j
) is dened on
fX 1
;:::X n
gnfX j
gbytheformula:
p(x 1
;:::x j 1
;x j+1
;:::x n
jjx^ j
)= n Y
i=1
i6=j p(x
i j pa
i
) (3:1)
withp(x i
jpa i
)asabove,but notingthatifX j
isaparentofX i
thenX
j takes
thevaluex^ j
.
This formula expressesthe eet of themanipulation do X
j = ^x
j
. A
manip-ulation of a CEG an be dened in an analogous manner by modifying the
distributionsofsomeoftherandomvariablessittingonpositions.
Denition1 Let(T;(T))be atree. LetD S(T)be asubset of the
situa-tionsof thetree,and ^ D
=f^(v 0
jv):v2D; v 0
2X(v)g beanew distribution
^
P(X(v)=v 0
)= (v
0
jv) v2=D
^
(v
0
jv) v2D
forallv 0
2X(v); v2S(T).
The manipulated tree is the tree so dened, and the manipulated CEG is the
CEGof the manipulatedtree.
Denition2 A manipulation of atree isalled positioned if the partition of
the positions after the manipulation isequal tooraoarsening ofthe partition
before manipulation. It is alled staged if the partition of the stages after the
manipulation isequaltoor aoarseningof the partition before manipulation.
A positioned manipulation of a tree treats all sample units identially when
their future development distributions are idential. A staged manipulation
treatssampleunits identially if theirnext developmentin theidle systemis
the same. In our experiene, it is usually suÆientto restrit study to
posi-tionedmanipulations, and note that all manipulations of aBN onsidered by
Pearl[17℄[18℄[19℄arebothpositionedandstaged.
This has a useful onsequene for manipulation of CEGs: As manipulations
tendtodestroysomeoftheonditional independenestrutureofamodelany
way, we anhoose to suppress those onditional independene properties
en-oded by oloured and undireted edges and treat our CEG as simple. For
simpleCEGs,eahpositionwisalsoastageu,andinterventionsinthelassof
positionedmanipulations ofatreeanbeenatedonaCEGsimplyby
repla-ing (T;(T)) by(C ;(C)) in Denition 1; D S(T) byD W(C)nfw
1 g;
^ D
=f^(v 0
jv):v2D; v 0
2X(v)gby ^ D
=f^(e(w;w 0
)jw):w2Dg,where
^
(e(w;w
0
)jw)isanewdistribution oftherandomvariable X(u)foru=w.
Thestandardmanipulations ofaBN arethosethat fore someomponentsof
the network to take preassigned values, as in expression (3.1). The analogue
for CEGs is to onsider manipulations whih fore all paths to pass through
aspeiedset of positions W. This ould be, for example, theassignmentof
patientswith partiularvaluesof aset ofovariates(detailed bytheirurrent
positions)to apartiulartreatmentregime (asetofsubsequentpositions W).
In a CEG, for a set of positions W, we let pa(W) = fw 2 W(C) :
9 w
0
2 W suh that e(w;w
0
) 2 E
d
(C)g be the set of positions whih have
an outgoing edge terminating in a position within W. We all the set W a
manipulationsetifallroot-to-sinkpathsinCpassthroughexatlyoneposition
inpa(W),and eahpositionin pa(W)hasexatlyonehildin W.
Example3.1 InExample2.1,onsider themanipulationforedtow
1
(manip-ulationset W =fw
1
g;pa(W)=fw 0
g),whih orresponds toensuringthat the
suspetgoestoourt.
Thisassignsaprobabilityof1totheedgee(w 0
;w 1
),andallvertiesandedges
notlyingonaw 0
!w
1
!w
1
inourmanipulatedCEGCareidentialtotheorrespondingedge-probabilities
inC exepttheprobabilityontheedgee(w 0
;w 1
). OurmanipulatedCEG
^ C is
giveninFigure3. Asallprobabilitiesafterthemanipulationremainunhanged,
wehavestagesasmarked.
1 (x
1
1
)
x
2
1
x
2
0
x
5
1
x
5
0
x
5
0
x
5
1
w
10
x
4
0
x
4
1
x
5
0
x
5
0
x
4
0
x
4
1
x
3
1
x
3
0
x
5
1
x
5
1
w
7
w
9
w
8
w
0
w
1
w
3
w
6
w
5
w
11
w
12
w
13
w
14
w
inf
x
6
1
x
6
0
x
6
0
x
4
0
x
4
1
x
6
0
x
6
1
Figure3: ManipulatedCEG
^
Cformanipulationto w 1
NotethatonaBN,weanspeify,forexample,thatapatientistotakesome
treatment, but weannot speifyhowweare goingto ensurethat thepatient
takesthis treatment. The CEG allows us more exibility { we an onsider
agreaterrange ofinterventions,many ofwhih may alterthe topologyof the
graphinmoreomplexwaysthanillustratedhere.
We assume that Figure 3 shows a CEG whih is valid for our manipulation,
butweneedtoexeriseareinmakingthisassumption. Ifajudge isavailable,
suÆientmoneyhasbeenstolen et., thenthepolie, believing S tobeguilty,
willmakeadeisiontoproeed. Inthisaseourmanipulated CEGisalmost
ertainlyvalid. ButsupposethepolieobtainCCTVfootageshowingS to be
present{thenthepoliewillagainmakeadeisiontoproeed(ensuringthere
is a judge available, and ignoring polie-fore poliy if neessary). This an
alsobeinterpretedasamanipulation tow 1
, butin thisaseedge-probabilities
downstreamofthemanipulationmaywellhange{thepreseneofSonCCTV
footage may inreasetheprobabilityof thewitnessidentifying S forexample.
Thismanipulation mayalsoalter thetopologyofthe manipulatedCEG {the
witnessfailingto identifyS maynolongerresultautomatiallyinanaqittal.
If we now onsider the manipulation fored to w
13
, we note that not all
0 2 4 13 1
in pa(w 13
). Thismanipulation none-the-lesshasastraight-forward
interpreta-tion(asaontingentmanipulation){ifthepolieproeed,thewitnessisfored
nottoidentify thesuspet. ACEGforthisinterpretationisgivenin Figure4.
x
1
1
x
2
1
x
2
0
x
2
0
x
2
1
x
3
0
xxxx
1 (x
5
0
)
1 (x
5
0
)
w
10
x
4
0
x
4
1
1 (x
5
0
)
1 (x
5
0
)
x
3
1
x
4
0
x
4
1
x
3
1
x
3
0
w
7
w
9
w
8
w
0
w
1
w
2
w
3
w
4
w
6
w
5
w
11
w
13
w
inf
x
6
0
x
4
0
x
4
1
Figure4: ManipulatedCEG
^
C formanipulation tow 13
Asthemanipulationdenitionusesthephraseif the polie proeed,thereisno
reasonhereforalteringtheprobabilitiesonthee(w 2
;w 13
)ande(w 2
;w 14
)edges,
andsothestagestrutureisasinFigure4. Notethatthismanipulationmight
beenatedbyanoutsidemanipulator,suhasthesuspet'sbrother!
Themanipulation foringtofw
12 ;w
14
gisonsideredinsetion5.
Example3.2 Auniversityhas residenebloksofapartments,withtworooms
eah. It alloates seond year students, either English (X 1
= 0) or Chinese
(X 1
= 1), to one of the two rooms in eah apartment. The seond room is
alloated toarst year student,either English (X 2
=0) or Chinese(X
2 =1),
andthis isdone atrandom. Asurveyhas reordedthatthe probability ofahigh
satisfationratingforstudentsplaedwithanotherstudentofthesameethniity
ishigherthanfor studentsplaedwith another studentof dierent ethniity.
Reording student satisfation via a binary indiator Y (Y = 1 being high
X
1
=
0
X
1
= 1
X
2
= 0
X
2
=
1
X
2
=
1
X
2
= 0
w
0
w
1
Y
=
1
Y
=
1
Y
=
0
Y
=
0
w
2
w
3
w
4
w
inf
Figure5: CEGforExample3.2
Theundiretededgebetweenw
1
and w
2
reetstherandomalloationofrst
year students to apartments. A possible BN for this problem, enoding the
independeneofX
1
andX
2
isgiveninFigure6(a).
X
1
X
2
X
2
Y
X
1
Y
(a)
(b)
Figure6: PossibleBNsforExample3.2
Ane utthroughthepositions fw
3 ;w
4
goftheCEGin Figure5givesus the
onditionalindependenepropertythatYq(X
1 ;X
2 )j
X
1 X
2
,apropertythat
annotbededuedfromtheBNinFigure6(a). Norisitpossibletodetermine
fromthisBN(orfrom thefatorisation oftheprobabilitymassfuntionof the
pathevents) whether thealloationof the seond year studentsours before
or after the alloationof the rst years. This property of the CEGallowsus
to onsider manipulations where, for example the university plaes rst year
3
thatthismanipulationwouldausetheremovaloftheundiretededgebetween
w 1
andw
2
, sineX 1
/
qX
2 j(X
1
=X
2 ).
Weould,ofourse,redeneourvariableX 2
sothatithadoutomespaef0;1g
orrespondingtofrst year studenthas sameethniity asseondyearstudent,
rstyearstudenthas dierentethniity fromseondyear studentg. Thiswould
giveus theBN as in Figure 6(b), and our manipulation would orrespond to
foringX
2
tothevalue0,withthedeletionofthearfromX 1
toX 2
. However,
thisnewBNdoesnotfullydesribetheidlesystem,asitnolongerenodesthe
propertythatrstyearstudentsarealloatedat random.
SoneitherBNisabletodesribeadequatelyboththeidleandthemanipulated
systems,whereastheCEGan.
4 Identifying the eets of manipulations
TherehasbeenonsiderablereentinterestinausalBNliterature[6℄[18℄[19℄in
studyingwhentheeetsofamanipulationonapre-speiedrandomvariable
Y an be identied from observing a subset of the BN's variables that are
observedor manifest intheidlesystem. Typially, suÆientonditionson the
topologyoftheBNaregivenforsuhidentiabilitytoexist. Thisallowsusto
designexperimentsontheidlesystemsoastobeabletoestimateeetsonthe
manipulatedsystem,forexampletheeetsofaproposednewtreatmentregime.
Thetopologyof theCEGanalso beused forthis purpose. Indeedit anbe
usedto nd funtionsof thedata (not just subsetsof possiblemeasurements)
thatwhenobservedintheidlesystemallowustoestimatetheeetofagiven
manipulationofaausalCEG.Asin[18℄weproveseveralsuÆientonditions
for identiability, and generalisePearl's Bak Door theorem to CEG models.
Werstneedtoprovidesomenotationandaoupleofdenitions.
Reallthatwindiates aposition inourCEG.Weuseto indiatea
root-to-sink(w 0
!w
1
)pathofourCEG.Eahisanatom ofthepath-algebraof
theCEG,and theset ofatomsisdenoted . A subpathofaroot-to-sinkpath
isdenotedormoreusually(w 1
;w 2
),wherew 1
andw
2
indiatethestartand
endpositionsofthesubpath.
A unionof atoms onstitutesan event,denoted ,and M is used to indiate
aunion of subpaths { usually this is of the form M(w 1
;w 2
) for positions w 1
and w
2
. (w) is used to representthe unionof allpaths passingthrough the
position w, and (e) theunionof allpaths passingthroughthe edgee. Both
(w)and (e) are events. ((w
1 ;w
2
))is the eventwhih is theunion ofall
pathsutilising thesubpath(w 1
;w 2
).
Weuse (w)=((w)) to denote the probability of passingthrough the
po-sitionw, whih isalso theprobabilityofreahingwfrom w 0
. Theprobability
of reahing w
2
from w
1
is ((w
2
) j (w
1
)), usually simplied to (w 2
j w 1
).
Similarly
(w 2
jw 1
)=(((w
1 ;w
2
))j (w
1
1 2 1
of as the probability of the subpath (w 1
;w 2
). By thinking of an edge as a
(veryshort)subpath,weanalsodene e
(w 2
jw 1
),theprobabilityoftheedge
e(w 1
;w 2
).
WenowonsiderrandomvariablesdenedonaCEG.
WeletY :!R bearandomvariable(measurable)withrespetto thepath
-algebra of the CEG; and let f
y
g be the partition of generated by Y {
namelyeah
y
istheunionofthose2forwhihY =y.
Denition3 A random variable Y isalledobservedif andonly ifindiators
ofthe events f y
gare observedfor all levelsy.
Denition4 Call a manipulation of a CEG (C ;(C)) fored to (the
position) wif:
1. itassignsprobability onetothe event(w),
2. all primitive probabilities in the manipulated CEG ( ^ C;
^ (
^
C)) assoiated
with edgesdownstreamof winC arethoseof the idlesystem.
InExample3.1,bothourmanipulationsaremanipulationsforedtoaposition.
Werstonsider aneetrandomvariable
^
Y dened onthepath-algebraof
^
C,wheretheinitialmanipulationsunderonsiderationaremanipulationsfored
tow. ^
Y generatesapartitionoftheroot-to-sinkpathsof ^
Cwitheahoutome
yorrespondingtoaunionofw 0
!w
1
paths
y .
Eah w
0
! w
1
path in ^
C anbe thought of as a onjuntion of a w
0
! w
subpath with a w ! w
1
subpath. We denote these subpaths by f(w
0 ;w)g
andf(w;w
1
)gandlettheunionofallw 0
!wsubpathsbeM(w
0 ;w).
Wewishto onsider
^
Y asperhapsameasurementof aneet aftera
manipu-lation foredto w, and sowewish ^
Y to be in somesense after ordownstream
of w. Todo this is straightforward. We requirethat our partition f y
g
on-sistsofeventseahofwhihisM(w 0
;w)onjoinedtoaunionofsubpathsfrom
f(w;w
1
)g{foroutomey,allthisunionM y
(w;w 1
).
We an dene a random variable Y on the path -algebra of C so that the
outomesof Y partitiontheroot-to-sinkpathsofC andwheneversuhapath
passesthroughwandtheequivalentpathin ^
Cbelongstotheevent ^
Y =ythen
inC thispathbelongsto theeventY =y.
Inpratialsituationsofourse,thisdeningofY and ^
Y isdonetheotherway
around. InaCEGofaBN,sets ofedgesthesamedistanefrom w 0
represent
outomesofthesamevariable, andwemight welllabelasubsetofsuhedges
withtheoutomey 0
forexample. TheeventY =y 0
wouldthenbetheunionof
allw 0
!w
1
pathsinCpassingthroughoneoftheseedges. Inthemanipulated
CEG ^
labelledy 0
,andtheeventY =y 0
willbetheunionofallw 0
!w
1
pathsin C
passingthroughoneoftheseedges.
Wherethere isnopossibilityofambiguityweandropthehatfrom ^ Y.
Lemma1 For alllevels y,under amanipulation foredtow
^ (
^
Y =y)=(Y =y j w)
provided thatin theunmanipulatedsystem(w)>0.
Wehavealreadyequatedtheevent ^
Y =ywiththeunionofw 0
!w
1
paths
y
in ^
C. Wean,withoutambiguity,whenworkingonC, equatetheeventY =y
withtheunionofw 0
!w
1
paths
y
in C, sinethosepathsthatompose
y
in ^
C are simplythose paths in C whih satisfyY = y and pass through the
position w. TheresultofLemma1anhenebeexpressedas:
^ (
y
)=(
y
j(w))
OneonsequeneofthisLemmaisthatforamanipulationforedtowitmaybe
possibletoobserveindiatorsontheeventsf y
\(w)gin theunmanipulated
systemandtoidentifytheeetonY ofthemanipulation,usingthisexpression.
Butthisisnothoweveralwayspossible,eveninmodelsthatanbedesribedby
aCBN.Whenweannotobservesuhindiators,weanoftenobserveindiators
forasetofoarserevents. Weshowbelowthatbeingabletoobserveindiators
on the events f
y
\(W)g (where W is some set of positions) an also be
suÆientforidentiability.
Denition5 A set of positions W of a CEG C is alled C-regular if no two
positionsinW lieon the samediretedpath of C.
We now onstrut an eet random variable assoiated with a manipulation
fored to W, where W is a C-regular set. So, as before, onsider a random
variable ^
Y denedonthepath-algebraof
^
C. Eahoutomeyof
^
Y orresponds
to a union of w 0
! w
1
paths in ^
C (
y
), and as before, we wish ^
Y to be
downstreamofW.
For a position w 2 W and outome y, we an speify an event M(w
0
;w)
M y
(w;w 1
) provided that the set f y
(w;w 1
)g is notempty. We then dene
ourevent ^
Y =y (or
y
)astheunionoverallw2W oftheeventsfM(w 0
;w)
M y
(w;w 1
)g. WedeneY onC asbefore, andwherethere isnopossibilityof
ambiguitywedropthehatfrom ^ Y.
Wewish to beable to stateonditionsfor theeet ofa manipulation fored
toaC-regularsetofpositionsW beingdeterminablediretlyfromprobabilities
in theidlesystem. Wedothis throughthe ideaof anamenable manipulation.
To aid us here, we onstrut agraph representingwhat happens up until we
reahagivenposition w. LetC
(w)denotetheolouredsubgraphofC whose
vertiesandedgesarethosealongthew 0
!wsubpathsofC,andwhose
edge-olouring(ie. edge-probabilities)isinheritedfromCalso. UsuallyC
aCEG.WriteK(C (w))forthesubsetofW(C)ofpositionsretainedinC (w),
withthe exeption ofw. Also,for aC-regular set ofpositions W, let C
(W)
denotetheolouredsubgraphofCwhosevertiesandedgesarethosealongthe
w 0
!w2W subpathsof C, andwhose edge-olouringisinheritedfrom C. If
W ontainsmorethanonepositionC
(W)isnotaCEG.Welet
K(C
(W))= [
w2W K(C
(w))
Denition6 Callasetof positionsW simpleifandonly if:
1. W isC-regular,
2. there exists a partition of the set K(C
(W)) into K
(C
(W)) and
K
(C
(W)), alled ative and bakground positions respetively, suh
that for w 2 W; ((w)) = ((M(w
0
;w))) an be deomposed as
A(w)B(w), whereA(w)isa funtionof the ativepositionsandB(w)
isafuntion ofthe bakgroundpositions,
3. A(w)=A(W)isonstant 8 w2W.
WhenW ontainsasingleposition,W islearlysimple.
Denition7 Amanipulation isalled amenableforingto aset W if:
1. the setW issimplein (C ;),
2. the setW issimplein ( ^ C;
^
), and
^
(W)=1,
3. (C)and
^ (
^
C)dier onlyon edges whoseparents liein K
(C
(W)).
Example4.1 Consider the binary BN and orresponding CEG in Figure 7.
Letthe set W =fw 7
;w 9
g.
Here C
(W) would bethe subgraphof theCEG in Figure 7onsisting of the
four subpaths joining w 0
to w 7
;w 8
; retaining the CEG's edge olouring (and
labels)but notitsundiretededges. Wehave
((w 7
))=(
0 )
X
d
(d) (x
0
jd)=(
0
)(x
0 )
((w 9
))=(
1 )
X
d
(d) (x
0
jd)=(
1
)(x
0 )
So here the position w 0
2 K
(C
(W)), B(w 7
) = (
0 ), B(w
9
) = (
1 ); the
positions w 3
;w 4
;w 5
;w 6
2K
(C
(W)),A(w 7
)=A(w
9
)=(x
0
)=A(W),and
w
0
w
1
w
2
w
3
w
inf
w
6
w
7
w
10
c
0
c
1
d
0
d
1
d
0
d
1
x
0
|d
0
x
1
|d
0
x
1
|d
1
x
0
|d
1
w
8
w
9
y
0
|c
0
x
0
w
4
w
5
y
0
|c
0
x
1
y
0
|c
1
x
0
y
0
|c
1
x
1
D
Y
C
X
Figure7: BNandCEGforExample4.1
IfwemanipulateC to W (equivalent tothe Pearlmanipulation do(X =x 0
)),
weget ^
C asinFigure8.
w
0
w
1
w
2
w
3
w
inf
w
6
w
7
c
0
c
1
d
0
d
1
d
0
d
1
1 (x
0
)
1
(x
0
)
1 (x
0
)
1
(x
0
)
w
9
y
0
|c
0
x
0
w
4
w
5
y
0
|c
1
x
0
Figure8: ^
^ ((w
7
))=(
0 )
X
d
(d)1=(
0
)1
^ ((w
9
))=(
1 )
X
d
(d)1=(
1
)1
andW issimplein ^ C.
^
diersonly ontheedges leavingw
3 ;w
4 ;w
5 ;w
6
,the edgesleavingtheative
positions; sothismanipulationis amenable.
Thepointofthesedenitionsisthat,inasensetobedenedbelow,therandom
variablesassoiatedwithpositionslyinginK
(C
(W))areindependentofthose
assoiatedwithpositionslyinginK
(C
(W)). Anamenablemanipulationmay
hange probabilitiesassoiatedwith variableslabelledbyativepositions, but
willalwaysleaveprobabilitiesassoiatedwithvariableslabelledbybakground
positions unhanged.
Lemma2 ConsideranamenablemanipulationforingtoasimplesetW. The
distribution of ^
Y (as dened above) is identied from the probabilities in the
unmanipulatedsystemofthe events fY =y;Wg,anditsprobabilitiesare given
bythe equation
^ (
^
Y =y)=
(Y =y;W)
(W)
where (W) =
P
w2W
((w)) =
P
w2W
((M(w 0
;w))), and provided that
((w))>08 w2W.
Moreformally
^ (
y
)=(
y
j(W))
where(W)=
S
w2W (w).
Notethat thevalidity oftheexpressionfor ^( y
)in Lemma 2dependson the
result^((w))=
((w))
((W))
holding,so hekingthis resultisasensiblestarting
pointinouranalysis.
Example4.2 The manipulationin Example4.1 isamenable.
Hereweandene
y
tobetheeventY =y 0
,andweget,fromFigure8that
^ (
y
)=(Y^ =y 0
)= X
()
X
d
(d)1(y
0 j;x
0 )
= X
()(y
0 j;x
( y
j(W))=(Y =y
0 jx 0 ) = P () P d (d)(x 0
j d)(y 0
j;x 0 ) P () P d (d)(x 0 jd) == X ()(y 0 j;x
0
)=^(Y =y 0 )=(^ y ) Notethat ((w7)) ((W)) = (0x0) (x 0 ) =( 0
) =^((w
7 ))
sineXqC (theneuts
throughfw
1 ;w
2
gandfw 3 ;w 4 ;w 5 ;w 6
ggiveus CqD; XqC jD ) XqC).
InaCBN,theeetofamanipulationofavariableX onalatervariableY an
beidentiedfrom observingthedistributionof theunmanipulatedpair(X;Y)
ifandonlyifthevetorof unobserved(hidden) variablesHin thesysteman
bepartitionedasH=(H 1 ;H 2 ),where H 2 q(H 1 ;X)
and
(Y;H 2
)qH
1 j X
It isstraightforwardto hekthat, foraCEG drawn in any order ompatible
withtheorderingofthevertiesofsuhaBN,theseareexatlytheonditions
ofLemma2. Inthisorrespondenethestatesofthevetorofhiddenvariables
H 1
andXdenethevaluestheativepositionstake,whilstthevetorofhidden
variablesH 2
denethevaluesthebakgroundpositionstake. SoLemma2isan
exatanalogueofthiswellknownresultforausalBNsforthemoregenerallass
ofCEGs. Moreover,theonditionsinLemma2onlydepend onanappropriate
fatorisationofprobabilitiesassoiatedwith themanipulated setW.
UsingPearl'sterminologyandthe(setsof)variablesX;Y;H 1
;H 2
wehavethat
(y jjx) =
X h 1 ;h 2 h (x;h 1 ;h 2 ;y)
(x jpa(x))
i
Notethat (X;H
1
) qH
2
)X qH
2
jH
1
,soweanequatePA (X)withH 1
,
andwrite
(y jjx)= X h1;h2 h (x;h 1 )(h 2 ;y jh
1 ;x)
(x jh
1 ) i = X h 1 ;h 2 h (h 1
)(xjh 1
)(h
2 ;y jx)
(xjh
1 )
i
using (Y;H
2
) q H
1 jX = X h 2 (h 2
;y jx)
ditioningX tox. NotethatinExample4.1weanlearlyseethat Cq(D;X)
and (Y;C)qD jX; soourativevariablesareDandX,andourbakground
variableisC.
Itisworthpointingoutthatifwedonotusethetwoonditionalindependene
statementsaswritten, butonlytheimpliations
(H 1
;X)qH 2
)XqH
2
jH
1
(Y;H 2
)qH
1
jX )Y qH
1
j(X;H 2
)
thentheonlyderivableexpressionis
(yjj x)= X
h2 (h
2
)(y jh 2
;x)
whihisofoursetheBakDoorformulaforBNs[18℄.
Example4.3 Note that if we break one of the onditions (eg. X q/ H
2 ) we
no longer have an amenable manipulation. So, onsider the binary BN and
orresponding CEG inFigure9.
w
0
w
1
w
2
w
3
w
inf
w
6
w
7
w
10
c
0
c
1
d
0
d
1
d
0
d
1
x
0
|c
0
d
0
x
1
|c
0
d
0
x
1
|c
0
d
1
x
0
|c
0
d
1
w
8
w
9
y
0
|c
0
x
0
w
4
w
5
y
0
|c
0
x
1
y
0
|c
1
x
0
y
0
|c
1
x
1
x
0
|c
1
d
0
x
1
|c
1
d
0
x
0
|c
1
d
1
x
1
|c
1
d
1
D
Y
C
X
Figure9: BNandCEGforExample4.3
IfweletthesetW =fw 7
;w 9
g,then
((w 7
))=(
0 )
X
d
(d)(x
0 j
0 ;d)
((w 9
))=(
1 )
X
d
(d)(x
0 j
ManipulatingtoW wegetthesame ^
C asbefore,so
^ (y
0 )=
X
()(y
0 j;x
0 )
butwenowhave
(y 0
jx 0
)= P
()
P
d
(d) (x
0
j;d) (y 0
j;x 0
) P
()
P
d
(d) (x
0 j;d)
whihnolongersimpliestothisexpression. Notethat here
(w 7
)
(W)
= (
0 ;x
0 )
(x 0
)
=(
0 jx
0
)6=(
0
) =(w^
7 )
5 A BakDoor theoremfor ChainEvent Graphs
AkeyomponentofausalanalysisonBNsisPearl'sBakDoortheorem[18℄[19℄,
whihowesitsderivationinparttotherealisationthatmanymanipulationsare
impossible,unethialorprohibitivelyexpensiveinpratie,ormaybepossible
toenatbutsomeoftheireetsmaybeimpossibletoobserve. TheBakDoor
theoremgivessuÆient onditionsforidentifying theeet onavariable Y of
manipulationof avariable X whenweare ableto observethevaluestakenby
onlyasubsetZ oftheremainingvariablesinthesystem. IfthesetZ ishosen
arefullyweanalulasteorestimatethiseetfromapartiallyobservedidle
system.
Inthis setionweproduean analogoustheorem that applies agraphialand
suÆient riterion to a CEG to determine whether wean identify the eet
ofanobservedmanipulationonarandomvariableY fromtheobservationofa
randomvariableZ (happeningbefore themanipulation in thepartialordering
induedbythepaths)intheunmanipulatedsystem. Theevent-basedtopology
oftheCEGallowsustoonsidernotonlyawiderlassofidlesystemmodels,but
alsoawiderlassofmanipulationsofthesethanispossiblewithaBN.Similarly,
ourrandomvariableZ nolongerneedstoorrespondtoanyxedsubsetofthe
measurement variables of the problem, givingus moreopportunity of nding
anappropriateprobabilityexpression.
WehavepreviouslyonsideredC
(w)andC
(W),graphsrepliatingthe
topol-ogy of C from w
0
to w or to a set of positions W. The graph C(w) whih
repliatesthe topology of C from w to w 1
is, unlike C
(w), automatially a
CEG,withwasroot-node. WeanalsoreateaCEGrepliatingthetopology
ofCfrom W tow
1 .
Denition8 For a set of C-regular positions W W(C), the graph C(W)
with vertexset V(C(W)), diretededge set E d
(C(W)) andundireted edge set
E u
1. V(C(W)) onsists of the union of fw 0
g, a new root-node, with the set
of preisely those positions from W(C) whih lie on a w! w 1
subpath
inC,for somew2W.
2. The root-node w 0
is onneted by an edge to eah w 2 W. E
d
(C(W))
onsists of the union of the set fe(w 0
;w)g w2W
with the set of preisely
those edges fromE d
(C) whih lie on a w!w
1
subpath in C, for some
w2W.
3. Edge-olourings(ie. edge-probabilities)onw!w 1
subpathsofC(W)(for
w2W) areretainedfrom C.
4. The edgee(w 0
;w)(w2W) isgiven theprobability
((w))
((W)) .
5. If twopositions inV(C(W))wereonnetedby an undiretededge in C,
thentheyareonnetedinC(W). E u
(C(W))isthesetofundiretededges
inC(W).
Itisstraightforwardtoshowthat C(W)isaCEG.
Lemma3 For sets of C-regular positions W
1 ;W
2
W(C), W
2
is simple
in the CEG C(W
1
) if and only if the probability ((w 2
) j (W
1
)) an be
deomposedas A(W
2
)B(w
2
)for all w 2
2W
2
(whereA(W
2
) isonstant for
allw 2
2W
2 ).
ThisisanimportantresultasitmeansthatwhetherasetW 2
issimpleinC(W 1
)
anbehekedonC,withouttheneessityofdrawingtheCEGC(W
1 ).
We now let Z be a random variable observed on C, whose events
fZ = zg f
z
g partition the set of w 0
! w
1
paths of C; and onsider
fw 1
g,a ne utof C suh that eah
z
is exatlytheset ofw 0
!w
1 paths
in C passing through a (speied) subset of positions from fw
1
g. We an
then, without ambiguity, identify eah event
z
with this set of positions {
sayfw 1 z
g.
Ifwelettheset of positions to whih weintendto manipulatebeW =fw 2
g,
then for Z to our before the manipulation we require that every position
w 2
2W liesonapathin C betweensomepositionw 1
2
z
(forsomelevelz)
andw
1
. Notealsothatasoursetfw 1
gisgoingtotaketheroleofZinourBak
Doortheorem,weneedittobetheasethatthemanipulationdoesnothange
anyprimitiveprobabilitiesfrom theidlesystemlyingonasubpathbetweenw 0
and the positions in fw 1
g. To ensurethis we need to stipulate that for eah
w 1
2fw
1
g, theremust exist aw 0 ! w 1 !w 2 !w 1
pathfor somew 2
2W
{if there existed w 1
2 fw
1
g forwhih there was nosuh w
2
, then ^((w
1 ))
would equalzero, and hene would notequal ((w
1
)). Having imposed this
ondition,weanensurethattheprobabilityofZ =z ( z
)isthesamein ^ Cas
C( z
)-measureableevents(foreahlevelz)suhthat
(Y =y)=
X
z
(Y =yjZ =z)(Z=z)
sinef
z
gpartitions thew 0
!w
1
pathsofC.
Denition9 A set of C-regular positions W W(C) isalled simple
ondi-tionedonZ if
1. W =
S
z W
z
where W
z
issimpleinC( z
).
2. ThereisadiretedpathinCfromeahpositionw 1 z
2
z
throughaposition
w 2
2W,andW
z
is thesetof preisely those positionsinW whih lieon
aw
0
!w
1 z
!w
1
path for somew 1 z
2
z .
Notethattheunionin item1isnotadisjointunion.
ConsideranamenablemanipulationtoasetW,andletW besimpleonditioned
onZ. ThenZ isalledaBak Doorvariable to themanipulation. Noteagain
thatsuhamanipulationdoesnothangeanyprimitiveprobabilitiesfrom the
idlesystemlying onasubpath betweenw 0
andpositions in z
. Letting ^
Y be
theimage of Y in themanipulated CEG, we havethat f
^
Y =y j Z =zgare
C( z
)-measureableeventssuhthat
^ (
^
Y =y)=
X
z ^ (
^
Y =yjZ =z)(Z^ =z)
Theorem 1 If a set W is simple onditioned on Z (a Bak Door variable),
then the distribution of Y after an amenable manipulation to W is identied
fromtheprobabilities (intheidlesystem)ofthe eventsfY =y;W;Z =zg,and
itsprobabilities aregiven by:
^ (
^
Y =y)=
X
z
(Y =y;W jZ =z)
(W j Z=z)
(Z=z)
ormore formally, as
^ (
y
)=
X
z (
y
j (W);
z
)(
z )
Itis worthstressingthat the partition f z
gis onstrutedso astohelp us to
alulate ^(
y
), andthat the hoie of positionswithin f
z
gwilltherefore
depend onthoseeventswhihareobservableormanifestwithin thesystem. If
aolletionofpositionswithinf z
gareindistinguishablethroughobservations
possible ontheidle system, then we would assign these positions to the same
z
