Graphical Models - svivek · • Example: Hidden Markov Model –Naïve Bayes classifier is also a simple Bayes net • Sometimes Bayes nets cannot represent the independence ...

CS6355:StructuredPrediction

GraphicalModels

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Sofar…Wediscussedsequencelabelingtasks:• HMM:HiddenMarkovModels• MEMM:MaximumEntropyMarkovModels• CRF:ConditionalRandomFieldsAllthesemodelsusealinearchainstructuretodescribetheinteractionsbetweenrandomvariables.

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HMM MEMM CRF

Thislecture

Graphicalmodels– Directed:BayesianNetworks– Undirected:MarkovNetworks(MarkovRandomField)

• Representations• Inference• Learning

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GraphicalModels

• Alanguagetorepresentprobabilitydistributionsovermultiplerandomvariables– Directedorundirectedgraphs

• Encodesconditionalindependenceassumptions• Orequivalently,encodesfactorizationofjointprobabilities.

• Generalmachinery for– Algorithmsforcomputingmarginalandconditionalprobabilities

• Recallthatwehavebeenlookingatmostprobablestatessofar• Exploitinggraphstructure

– An“inferenceengine”

– Canintroducepriorprobabilitydistributions• Becauseparametersarealsorandomvariables

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Decomposejointprobabilityviaadirectedacyclicgraph– Nodesrepresentrandomvariables– Edgesrepresentconditionaldependencies– Eachnodeisassociatedwithaconditionalprobabilitytable

BayesianNetwork

ExamplefromRussellandNorvig 5

Decomposejointprobabilityviaadirectedacyclicgraph– Nodesrepresentrandomvariables– Edgesrepresentconditionaldependencies– Eachnodeisassociatedwithaconditionalprobabilitytable

BayesianNetwork

ExamplefromRussellandNorvig

P(B,E,A,J,M)=P(B)P(E)P(A|B,E)P(J|A)P)M|A)

Compactrepresentationofthejointprobabilitydistribution

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• Localindependencies:Anodeisindependentwithitsnon-descendantsgivenitsparents.

• Topologicalindependencies:Anodeisindependentofallothernodesgivenitsparents,childrenandchildren’sparents—thatisgivenitsMarkovBlanket.

• Globalindependencies:D-separation

IndependenceAssumptionsofaBN

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(Xi ? NonDescendants(Xi)|Parents(Xi))

(Xi ? Xj |MB(Xi)) for all j 6= i

ExamplefromDaphneKoller

• Localindependencies:Anodeisindependentwithitsnon-descendantsgivenitsparents.

• Topologicalindependencies:Anodeisindependentofallothernodesgivenitsparents,childrenandchildren’sparents—thatisgivenitsMarkovBlanket.

• Globalindependencies:D-separation

IndependenceAssumptionsofaBN

Wheredotheindependenceassumptionscomefrom?

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(Xi ? NonDescendants(Xi)|Parents(Xi))

(Xi ? Xj |MB(Xi)) for all j 6= i

ExamplefromDaphneKoller

• Localindependencies:Anodeisindependentwithitsnon-descendantsgivenitsparents.

• Topologicalindependencies:Anodeisindependentofallothernodesgivenitsparents,childrenandchildren’sparents—thatisgivenitsMarkovBlanket.

• Globalindependencies:D-separation

IndependenceAssumptionsofaBN

Wheredotheindependenceassumptionscomefrom?

Domainknowledge

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(Xi ? NonDescendants(Xi)|Parents(Xi))

(Xi ? Xj |MB(Xi)) for all j 6= i

ExamplefromDaphneKoller

BayesianNetwork

• Example:HiddenMarkovModel– NaïveBayesclassifierisalsoasimpleBayesnet

• SometimesBayesnetscannotrepresenttheindependencerelationswewantconveniently.– Eg:Segmentinganimagebyassigningalabeltoeachpixel

• Say,wewantadjacentlabelstoinfluenceeachotherTwoproblems:1. Whatistherightdirectionofarrows?

2. Foranychoiceofthearrows,strangedependenciesshowup.X8 isindependentofeverythinggivenitsMarkovblanket(othercirclednodeshere)

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BayesianNetwork

• Example:HiddenMarkovModel– NaïveBayesclassifierisalsoasimpleBayesnet

• SometimesBayesnetscannotrepresenttheindependencerelationswewantconveniently.– Eg:Segmentinganimagebyassigningalabeltoeachpixel

• Say,wewantadjacentlabelstoinfluenceeachother

ExamplefromKevinMurphy 11

BayesianNetwork

• Example:HiddenMarkovModel– NaïveBayesclassifierisalsoasimpleBayesnet

• SometimesBayesnetscannotrepresenttheindependencerelationswewantconveniently.– Eg:Segmentinganimagebyassigningalabeltoeachpixel

• Say,wewantadjacentlabelstoinfluenceeachother

ExamplefromKevinMurphy

Twoproblems:1. Whatisthecorrectdirectionofarrows?

2. Foranychoiceofthearrows,strangedependenciesshowup.X8 isindependentofeverythinggivenitsMarkovblanket(othercirclednodeshere)

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FromdirectedtoundirectedNetworks

SometimesBayesnetscannotrepresenttheindependencerelationswewantconveniently.

– Eg:Segmentinganimagebyassigningalabeltoeachpixel• Say,wewantadjacentlabelstoinfluenceeachother

ExamplefromKevinMurphy 13

UndirectedGraphicalModels

• Anotherwayofdefiningconditionalindependence

• Generalstructure– Nodesarerandomvariables– Edges(hyper-edges)definedependencies

• Thenodesinacomplete subgraph formaclique.

a.k.a MarkovRandomFields/MarkovNetworks

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8cliques:{A},{B},{C},{D}{AB},{BC},{CD},{AD}

UndirectedGraphicalModels

• Anotherwayofdefiningconditionalindependence

• Generalstructure– Nodesarerandomvariables– Edges(hyper-edges)definedependencies

• Thenodesinacomplete subgraph formaclique.

a.k.a MarkovRandomFields/MarkovNetworks

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P (x) =1

Z

Y

c2Cliques

fc(xc)

8cliques:{A},{B},{C},{D}{AB},{BC},{CD},{AD}

P (A,B,C,D) =1

Zf1(A,B)f2(B,C)

f3(C,D)f4(A,D)

ThisisaGibbsdistributionifallfactorsarepositive

UndirectedGraphicalModels

• Anotherwayofdefiningconditionalindependence

• Generalstructure– Nodesarerandomvariables– Edges(hyper-edges)definedependencies

• Thenodesinacomplete subgraph formaclique.

a.k.a MarkovRandomFields/MarkovNetworks

Thejointprobabilitydecouplesovercliques.Everycliquexcassociatedwithapotential function f(xc)

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P (x) =1

Z

Y

c2Cliques

fc(xc)

8cliques:{A},{B},{C},{D}{AB},{BC},{CD},{AD}

P (A,B,C,D) =1

Zf1(A,B)f2(B,C)

f3(C,D)f4(A,D)

ThisisaGibbsdistributionifallfactorsarepositive

• Localindependencies:Anodeisindependentofallothernodesgivenitsneighbors.

• Globalindependencies:IfX,Y,Zaresetsofnodes,XisconditionallyindependentofYgivenZifremovingallnodesofCremovesallpathsfromAtoB

IndependenceAssumptionsofaMRF

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IndependenceAssumptionsofaMRF

Wheredotheindependenceassumptionscomefrom?

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• Localindependencies:Anodeisindependentofallothernodesgivenitsneighbors.

• Globalindependencies:IfX,Y,Zaresetsofnodes,XisconditionallyindependentofYgivenZifremovingallnodesofCremovesallpathsfromAtoB

IndependenceAssumptionsofaMRF

Wheredotheindependenceassumptionscomefrom?

Domainknowledge

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• Localindependencies:Anodeisindependentofallothernodesgivenitsneighbors.

• Globalindependencies:IfX,Y,Zaresetsofnodes,XisconditionallyindependentofYgivenZifremovingallnodesofCremovesallpathsfromAtoB

MRFtoFactorgraph

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Z: Calledthepartitionfunction, sumoverallassignmentstotherandomvariables

Normalize:

where

f(xc,µ)isoftenwrittenasexp(µT xc)Log-linearmodel

Factorgraphs

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Z: Calledthepartitionfunction,sumoverallassignmentstotherandomvariables

Normalize:

where

Factorgraph:Makesthefactorizationexplicit,factors insteadofcliques

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f(xc,µ)isoftenwrittenasexp(µT xc)Log-linearmodel

Whichcliques?

Factorgraphs

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Z: Calledthepartitionfunction,sumoverallassignmentstotherandomvariables

Normalize:

where

Factorgraph:Makesthefactorizationexplicit,factors insteadofcliques

Factors

Factors

Factors

f(xc,µ)isoftenwrittenasexp(µT xc)Log-linearmodel

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Factorgraphs

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Z: Calledthepartitionfunction,sumoverallassignmentstotherandomvariables

Normalize:

where

Factorgraph:Makesthefactorizationexplicit,factors insteadofcliques

Factors

Factors

Factors

f(xc,µ)isoftenwrittenasexp(µT xc)Log-linearmodel

P (x) =1

Z

fa(x1, x2, x4)fb(x2, x3, x5)fc(x4, x5)

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CommentsaboutMRFs

• Connectiontostatisticalphysics– IdenticaltoBoltzmanndistributioninenergybasedmodels– Probabilityofasystemexistinginastate:

• Ifx isdependentonallitsneighbors:– Ifxcanbeinoneoftwostates(binary),Ising model– Ifxcanbeinoneofmorethantwostates(multiclass),Pottsmodel

Z: Zustandssumme,“sumoverstates”,morecommonlycalledthepartitionfunction

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CommentsaboutMRFs

• Connectiontostatisticalphysics– IdenticaltoBoltzmanndistributioninenergybasedmodels– Probabilityofasystemexistinginastate:

• Ifx isdependentonallitsneighbors:– Ifxcanbeinoneoftwostates(binary),Ising model– Ifxcanbeinoneofmorethantwostates(multiclass),Pottsmodel

Energyofcliquecexistinginstatexc

Z: Zustandssumme,“sumoverstates”,morecommonlycalledthepartitionfunction

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CommentsaboutMRFs

• Connectiontostatisticalphysics– IdenticaltoBoltzmanndistributioninenergybasedmodels– Probabilityofasystemexistinginastate:

• Ifx isdependentonallitsneighbors:– Ifxcanbeinoneoftwostates(binary),Ising model– Ifxcanbeinoneofmorethantwostates(multiclass),Pottsmodel

Energyofcliquecexistinginstatexc

Z: Zustandssumme,“sumoverstates”,morecommonlycalledthepartitionfunction

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BayesianNetworksvs.MarkovNetworks

• Bothnetworksrepresent– Asetofconditionalindependencerelations– i.e,askeletonthatshowshowajointprobabilitydistributionis

factorized

• Bothnetworkshavetheoremsaboutequivalencebetweenconditionalindependenceand jointprobabilityfactorization

• GivenaBayesnet,thereisnotalwaysanequivalentMarkovrandomfield,andviceversa– Withsomecaveats– SeethechapteronundirectedgraphicalmodelsinKoller and

Friedman’sbook

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Inferenceingraphicalmodels

Ingeneral,computeprobabilityofasubsetofstates– P(xA),forsomesubsetsofrandomvariablesxA

• Note:Sofar,wehavegenerallyconsideredthesomethinglikeargmaxx P(x)

• Exactinference• “Approximate”inference

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1 2 3

4 5(moreonthisinfutureclasses)

Inferenceingraphicalmodels

Ingeneral,computeprobabilityofasubsetofstates– P(xA),forsomesubsetsofrandomvariablesxA

• Note:Sofar,wehavegenerallyconsideredtheequivalentofargmaxx P(x)

• Exactinference• “Approximate”inference

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1 2 3

4 5(moreonthisinfutureclasses)

Inferenceingraphicalmodels

Ingeneral,computeprobabilityofasubsetofstates– P(xA),forsomesubsetsofrandomvariablesxA

• Exactinference– Variableelimination

• Marginalizebysummingoutvariablesina“good”order• ThinkaboutwhatwedidforViterbi

– Beliefpropagation(exactonlyforgraphswithoutloops)• Nodespassmessagestoeachotherabouttheirestimateofwhattheneighbor’sstateshouldbe

– Generallyefficientfortrees,sequences(andmaybeothergraphstoo)

• “Approximate”inference30

1 2 3

4 5(moreonthisinfuturelectures)

Whatmakesanorderinggood?

Inferenceingraphicalmodels

Ingeneral,computeprobabilityofasubsetofstates– P(xA),forsomesubsetsofrandomvariablesxA

• Exactinference• “Approximate”inference– MarkovChainMonteCarlo

• GibbsSampling/Metropolis-Hastings– Variational algorithms

• Frameinferenceasanoptimizationproblem,perturbittoanapproximateoneandsolvetheapproximateproblem

– LoopyBeliefpropagation• RunBPandhopeitworks!

– Thenot-so-goodnews:Approximateinferenceisalsointractable!31

1 2 3

4 5

NP-hardingeneral,worksforsimplegraphs

(moreonthisinfuturelectures)