InferenceinBayesiannetworks

Chapter14.4–5

Chapter14.4–51

Outline

♦Exactinferencebyenumeration

♦Exactinferencebyvariableelimination

♦Approximateinferencebystochasticsimulation

♦ApproximateinferencebyMarkovchainMonteCarlo

Chapter14.4–52

Inferencetasks

Simplequeries:computeposteriormarginalP(Xi|E=e)e.g.,P(NoGas|Gauge=empty,Lights=on,Starts=false)

Conjunctivequeries:P(Xi,Xj|E=e)=P(Xi|E=e)P(Xj|Xi,E=e)

Optimaldecisions:decisionnetworksincludeutilityinformation;probabilisticinferencerequiredforP(outcome|action,evidence)

Valueofinformation:whichevidencetoseeknext?

Sensitivityanalysis:whichprobabilityvaluesaremostcritical?

Explanation:whydoIneedanewstartermotor?

Chapter14.4–53

Inferencebyenumeration

Slightlyintelligentwaytosumoutvariablesfromthejointwithoutactuallyconstructingitsexplicitrepresentation

Simplequeryontheburglarynetwork:BE

J

A

M

P(B|j,m)=P(B,j,m)/P(j,m)=αP(B,j,m)=αΣeΣaP(B,e,a,j,m)

RewritefulljointentriesusingproductofCPTentries:P(B|j,m)=αΣeΣaP(B)P(e)P(a|B,e)P(j|a)P(m|a)=αP(B)ΣeP(e)ΣaP(a|B,e)P(j|a)P(m|a)

Recursivedepth-firstenumeration:O(n)space,O(dn)time

Chapter14.4–54

Enumerationalgorithm

functionEnumeration-Ask(X,e,bn)returnsadistributionoverX

inputs:X,thequeryvariable

e,observedvaluesforvariablesE

bn,aBayesiannetworkwithvariables{X}∪E∪Y

Q(X)←adistributionoverX,initiallyempty

foreachvaluexiofXdo

extendewithvaluexiforX

Q(xi)←Enumerate-All(Vars[bn],e)

returnNormalize(Q(X))

functionEnumerate-All(vars,e)returnsarealnumber

ifEmpty?(vars)thenreturn1.0

Y←First(vars)

ifYhasvalueyine

thenreturnP(y|Pa(Y))×Enumerate-All(Rest(vars),e)

elsereturn∑

yP(y|Pa(Y))×Enumerate-All(Rest(vars),ey)

whereeyiseextendedwithY=y

Chapter14.4–55

Evaluationtree

P(j|a).90

P(m|a).70.01

P(m| a)

.05P(j| a)P(j|a)

.90

P(m|a).70.01

P(m| a)

.05P(j| a)

P(b).001

P(e).002

P( e).998

P(a|b,e).95.06

P( a|b, e).05P( a|b,e)

.94P(a|b, e)

Enumerationisinefficient:repeatedcomputatione.g.,computesP(j|a)P(m|a)foreachvalueofe

Chapter14.4–56

Inferencebyvariableelimination

Variableelimination:carryoutsummationsright-to-left,storingintermediateresults(factors)toavoidrecomputation

P(B|j,m)=αP(B) ︸︷︷︸

B

ΣeP(e) ︸︷︷︸

E

ΣaP(a|B,e) ︸︷︷︸

A

P(j|a) ︸︷︷︸

J

P(m|a) ︸︷︷︸

M

=αP(B)ΣeP(e)ΣaP(a|B,e)P(j|a)fM(a)=αP(B)ΣeP(e)ΣaP(a|B,e)fJ(a)fM(a)=αP(B)ΣeP(e)ΣafA(a,b,e)fJ(a)fM(a)=αP(B)ΣeP(e)fAJM(b,e)(sumoutA)=αP(B)fEAJM(b)(sumoutE)=αfB(b)×fEAJM(b)

Chapter14.4–57

Variableelimination:Basicoperations

Summingoutavariablefromaproductoffactors:moveanyconstantfactorsoutsidethesummationaddupsubmatricesinpointwiseproductofremainingfactors

Σxf1×···×fk=f1×···×fiΣxfi+1×···×fk=f1×···×fi×fX

assumingf1,...,fidonotdependonX

Pointwiseproductoffactorsf1andf2:f1(x1,...,xj,y1,...,yk)×f2(y1,...,yk,z1,...,zl)

=f(x1,...,xj,y1,...,yk,z1,...,zl)E.g.,f1(a,b)×f2(b,c)=f(a,b,c)

Chapter14.4–58

Variableeliminationalgorithm

functionElimination-Ask(X,e,bn)returnsadistributionoverX

inputs:X,thequeryvariable

e,evidencespecifiedasanevent

bn,abeliefnetworkspecifyingjointdistributionP(X1,...,Xn)

factors←[];vars←Reverse(Vars[bn])

foreachvarinvarsdo

factors←[Make-Factor(var,e)|factors]

ifvarisahiddenvariablethenfactors←Sum-Out(var,factors)

returnNormalize(Pointwise-Product(factors))

Chapter14.4–59

Irrelevantvariables

ConsiderthequeryP(JohnCalls|Burglary=true)BE

J

A

M

P(J|b)=αP(b)∑

eP(e)

∑

aP(a|b,e)P(J|a)

∑

mP(m|a)

Sumovermisidentically1;Misirrelevanttothequery

Thm1:YisirrelevantunlessY∈Ancestors({X}∪E)

Here,X=JohnCalls,E={Burglary},andAncestors({X}∪E)={Alarm,Earthquake}soMaryCallsisirrelevant

(ComparethistobackwardchainingfromthequeryinHornclauseKBs)

Chapter14.4–510

Irrelevantvariablescontd.

Defn:moralgraphofBayesnet:marryallparentsanddroparrows

Defn:Aism-separatedfromBbyCiffseparatedbyCinthemoralgraph

Thm2:Yisirrelevantifm-separatedfromXbyEBE

J

A

M

ForP(JohnCalls|Alarm=true),bothBurglaryandEarthquakeareirrelevant

Chapter14.4–511

Complexityofexactinference

Singlyconnectednetworks(orpolytrees):–anytwonodesareconnectedbyatmostone(undirected)path–timeandspacecostofvariableeliminationareO(d

kn)

Multiplyconnectednetworks:–canreduce3SATtoexactinference⇒NP-hard–equivalenttocounting3SATmodels⇒#P-complete

ABCD

123

AND

0.50.5 0.5 0.5

L L

LL

1. A v B v C

2. C v D v A

3. B v C v D

Chapter14.4–512

Inferencebystochasticsimulation

Basicidea:1)DrawNsamplesfromasamplingdistributionS

Coin

0.5 2)ComputeanapproximateposteriorprobabilityP3)ShowthisconvergestothetrueprobabilityP

Outline:–Samplingfromanemptynetwork–Rejectionsampling:rejectsamplesdisagreeingwithevidence–Likelihoodweighting:useevidencetoweightsamples–MarkovchainMonteCarlo(MCMC):samplefromastochasticprocess

whosestationarydistributionisthetrueposterior

Chapter14.4–513

Samplingfromanemptynetwork

functionPrior-Sample(bn)returnsaneventsampledfrombn

inputs:bn,abeliefnetworkspecifyingjointdistributionP(X1,...,Xn)

x←aneventwithnelements

fori=1tondo

xi←arandomsamplefromP(Xi|parents(Xi))

giventhevaluesofParents(Xi)inx

returnx

Chapter14.4–514

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–515

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–516

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–517

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–518

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–519

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–520

Example

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

Chapter14.4–521

Samplingfromanemptynetworkcontd.

ProbabilitythatPriorSamplegeneratesaparticulareventSPS(x1...xn)=Π

ni=1P(xi|parents(Xi))=P(x1...xn)

i.e.,thetruepriorprobability

E.g.,SPS(t,f,t,t)=0.5×0.9×0.8×0.9=0.324=P(t,f,t,t)

LetNPS(x1...xn)bethenumberofsamplesgeneratedforeventx1,...,xn

Thenwehave

limN→∞

P(x1,...,xn)=limN→∞

NPS(x1,...,xn)/N

=SPS(x1,...,xn)

=P(x1...xn)

Thatis,estimatesderivedfromPriorSampleareconsistent

Shorthand:P(x1,...,xn)≈P(x1...xn)

Chapter14.4–522

Rejectionsampling

P(X|e)estimatedfromsamplesagreeingwithe

functionRejection-Sampling(X,e,bn,N)returnsanestimateofP(X|e)

localvariables:N,avectorofcountsoverX,initiallyzero

forj=1toNdo

x←Prior-Sample(bn)

ifxisconsistentwithethen

N[x]←N[x]+1wherexisthevalueofXinx

returnNormalize(N[X])

E.g.,estimateP(Rain|Sprinkler=true)using100samples27sampleshaveSprinkler=true

Ofthese,8haveRain=trueand19haveRain=false.

P(Rain|Sprinkler=true)=Normalize(〈8,19〉)=〈0.296,0.704〉

Similartoabasicreal-worldempiricalestimationprocedure

Chapter14.4–523

Analysisofrejectionsampling

P(X|e)=αNPS(X,e)(algorithmdefn.)=NPS(X,e)/NPS(e)(normalizedbyNPS(e))≈P(X,e)/P(e)(propertyofPriorSample)=P(X|e)(defn.ofconditionalprobability)

Hencerejectionsamplingreturnsconsistentposteriorestimates

Problem:hopelesslyexpensiveifP(e)issmall

P(e)dropsoffexponentiallywithnumberofevidencevariables!

Chapter14.4–524

Likelihoodweighting

Idea:fixevidencevariables,sampleonlynonevidencevariables,andweighteachsamplebythelikelihooditaccordstheevidence

functionLikelihood-Weighting(X,e,bn,N)returnsanestimateofP(X|e)

localvariables:W,avectorofweightedcountsoverX,initiallyzero

forj=1toNdo

x,w←Weighted-Sample(bn)

W[x]←W[x]+wwherexisthevalueofXinx

returnNormalize(W[X])

functionWeighted-Sample(bn,e)returnsaneventandaweight

x←aneventwithnelements;w←1

fori=1tondo

ifXihasavaluexiine

thenw←w×P(Xi=xi|parents(Xi))

elsexi←arandomsamplefromP(Xi|parents(Xi))

returnx,w

Chapter14.4–525

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0

Chapter14.4–526

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0

Chapter14.4–527

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0

Chapter14.4–528

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0×0.1

Chapter14.4–529

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0×0.1

Chapter14.4–530

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0×0.1

Chapter14.4–531

Likelihoodweightingexample

Cloudy

Rain Sprinkler

WetGrass

C

TF

.80

.20

P(R|C) C

TF

.10

.50

P(S|C)

SR

TTTFFTFF

.90

.90

.99

P(W|S,R)

P(C).50

.01

w=1.0×0.1×0.99=0.099

Chapter14.4–532

Likelihoodweightinganalysis

SamplingprobabilityforWeightedSampleis

SWS(z,e)=Πli=1P(zi|parents(Zi))

Note:paysattentiontoevidenceinancestorsonlyCloudy

Rain Sprinkler

WetGrass

⇒somewhere“inbetween”priorandposteriordistribution

Weightforagivensamplez,eisw(z,e)=Π

mi=1P(ei|parents(Ei))

WeightedsamplingprobabilityisSWS(z,e)w(z,e)

=Πli=1P(zi|parents(Zi))Π

mi=1P(ei|parents(Ei))

=P(z,e)(bystandardglobalsemanticsofnetwork)

Hencelikelihoodweightingreturnsconsistentestimatesbutperformancestilldegradeswithmanyevidencevariablesbecauseafewsampleshavenearlyallthetotalweight

Chapter14.4–533

ApproximateinferenceusingMCMC

“State”ofnetwork=currentassignmenttoallvariables.

GeneratenextstatebysamplingonevariablegivenMarkovblanketSampleeachvariableinturn,keepingevidencefixed

functionMCMC-Ask(X,e,bn,N)returnsanestimateofP(X|e)

localvariables:N[X],avectorofcountsoverX,initiallyzero

Z,thenonevidencevariablesinbn

x,thecurrentstateofthenetwork,initiallycopiedfrome

initializexwithrandomvaluesforthevariablesinY

forj=1toNdo

foreachZiinZdo

samplethevalueofZiinxfromP(Zi|mb(Zi))

giventhevaluesofMB(Zi)inx

N[x]←N[x]+1wherexisthevalueofXinx

returnNormalize(N[X])

Canalsochooseavariabletosampleatrandomeachtime

Chapter14.4–534

TheMarkovchain

WithSprinkler=true,WetGrass=true,therearefourstates:

Cloudy

Rain Sprinkler

WetGrass

Cloudy

Rain Sprinkler

WetGrass

Cloudy

Rain Sprinkler

WetGrass

Cloudy

Rain Sprinkler

WetGrass

Wanderaboutforawhile,averagewhatyousee

Chapter14.4–535

MCMCexamplecontd.

EstimateP(Rain|Sprinkler=true,WetGrass=true)

SampleCloudyorRaingivenitsMarkovblanket,repeat.CountnumberoftimesRainistrueandfalseinthesamples.

E.g.,visit100states31haveRain=true,69haveRain=false

P(Rain|Sprinkler=true,WetGrass=true)=Normalize(〈31,69〉)=〈0.31,0.69〉

Theorem:chainapproachesstationarydistribution:long-runfractionoftimespentineachstateisexactlyproportionaltoitsposteriorprobability

Chapter14.4–536

Markovblanketsampling

MarkovblanketofCloudyisCloudy

Rain Sprinkler

WetGrass

SprinklerandRainMarkovblanketofRainis

Cloudy,Sprinkler,andWetGrass

ProbabilitygiventheMarkovblanketiscalculatedasfollows:P(x

′i|mb(Xi))=P(x

′i|parents(Xi))ΠZj∈Children(Xi)P(zj|parents(Zj))

Easilyimplementedinmessage-passingparallelsystems,brains

Maincomputationalproblems:1)Difficulttotellifconvergencehasbeenachieved2)CanbewastefulifMarkovblanketislarge:

P(Xi|mb(Xi))won’tchangemuch(lawoflargenumbers)

Chapter14.4–537

Summary

Exactinferencebyvariableelimination:–polytimeonpolytrees,NP-hardongeneralgraphs–space=time,verysensitivetotopology

ApproximateinferencebyLW,MCMC:–LWdoespoorlywhenthereislotsof(downstream)evidence–LW,MCMCgenerallyinsensitivetotopology–Convergencecanbeveryslowwithprobabilitiescloseto1or0–Canhandlearbitrarycombinationsofdiscreteandcontinuousvariables

Chapter14.4–538