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CS839:ProbabilisticGraphicalModels
Lecture1:IntroductiontoGraphicalModels
TheoRekatsinas
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Acknowledgement:adaptedslidesbyEricXing
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1.Introduction,admin&setup
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WhoamI…
Instructor(me)TheoRekatsinas• FacultyintheComputerSciencesandpartoftheUW-DatabaseGroup• Research:dataintegrationandcleaning,statisticalanalytics,andmachinelearning.• [email protected]• Officehours:Byappointment@CS4361
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CourseWebpage:
https://thodrek.github.io/CS839_fall18/
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Logistics
• Textbooks:• ProbabilisticGraphicalModels,byDaphneKoller andNir Friedman• IntroductiontoStatisticalRelationalLearning,byLise Getoor andBenTaskar
• Officehours:• Byappointment.Justsendmeanemail
• Homeworksubmission:• WewilluseCanvas
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AssignmentsandGradingLogistics
• 3homeworkassignments:20%ofgrade• Theoryexercises,implementationexercises
• Midterm:30%ofgrade• Inclassexam• ~weak#9
• Finalproject:50%ofgrade• Projectproposal:10%ofgrade(~weak#9)• Proposalpresentation:10%ofgrade• Finalreport:30%ofgrade(dueonDec20th)• Ingroupsof(upto)3.Ideallyitshouldbethree.Groupsshouldbeformedinthefirsttwoweeks.
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Projectexamples
• ApplyingPGMtothedevelopmentofareal,substantialMLsystem• Buildaweb-scalefakenewsdetector.• Buildastorylinetrackingsystemfornewsmedia.• Designandimplementthestate-of-the-artknowledgebaseemgeddings
• Theoryand/oralgorithmicprojects• Amoreefficientapproximateinferencealgorithm.• Whenisinferenceinthepresenceofnoisyobservationshard?• WhencanweapproximatePGMswithfeed-forwardnetworks?
• System’s• ImplementMarkovlogicontopofPyro
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2.Classoverview
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Whataregraphicalmodels?
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MGraph Model
D ⌘ {X(i)1 , X(i)
2 , . . . , X(i)m }NI=1
Data
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PGMsallowustoreasonaboutuncertainty
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InformationExtraction`
DataCleaningWeakSupervision
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FundamentalQuestions
• Representation• Howtocapture/modeluncertaintiesinpossibleworlds?• Howtoencodeourdomainknowledge/assumptions/constraints?
• Example:IsyourGradeindependentoftheDifficultyoftheclass?
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Difficulty Intelligence Grade
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FundamentalQuestions
• Inference• Howdoweanswerquestions/queriesaccordingtothemodelinhandandtheavailabledataP(X|Data)
• Example:WhatwillyourGradebeifDifficultyis“high”?
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Difficulty Intelligence Grade
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FundamentalQuestions
• Learning• Whatmodelis“right”forthedata?
• Example:Whatifwehave(Difficulty=“Low”,Intelligence=“High”,Grade=“High”)forperson1,(Difficulty=“High”,Intelligence=“High”,Grade=“Low”)forperson2,etc?
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Difficulty Intelligence Grade
M = arg max
M2MF (D;M)
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BasicProbabilityConcepts
• Representation:Whatisthejointprobabilitydistributiononmultiplevariables?
• Howmanystateconfigurationswehaveintotal?• Aretheyallneededtoberepresented?• Whatinsightsdowegetfromthismodel?
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P (X1, X2, X3, X4, X5, X6, X7, X8)
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BasicProbabilityConcepts
• Learning:Wheredowegetalltheseprobabilities?
• Maximal-likelihoodestimation?Howmanydatadoweneed?• Arethereotherestimationprinciples?• Wheredoweputdomainknowledgeintermsofplausiblerelationshipsbetweenvariables,andplausiblevaluesoftheprobabilities?
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BasicProbabilityConcepts
• Inference:Ifnotallvariablesareobservable,howtocomputetheconditionaldistributionoflatentvariablesgivenevidence?
• Assumeisgiven.Computingrequiressummingoverallconfigurationsoftheunobservedvaribles.
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X1 P (X2|X1)26
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Whatisagraphicalmodel?
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AMultivariateDistributioninHigh-DSpace
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Example:Apossibleworldforcellularsignaltransduction
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Signaltransduction istheprocessbywhichachemicalorphysicalsignalistransmittedthroughacellasaseriesofmolecularevents
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Example:Apossibleworldforcellularsignaltransduction
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StructureSimplifiesRepresentation
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Arrowsindicatedependenciesamongstvariables
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ProbabilisticGraphicalModels
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• If areconditionallyindependent(asdescribedbyaPGM),thejointcanbefactoredtoaproductofsimplerterms,e.g.,
• So,whyaPGM?Wecanincorporatedomainknowledge• 1+1+2+2+2+4+2+4=18,a16-foldreductionfrom28inrepresentationcost!
Xi’s
P (X1, X2, X3, X4, X5, X6, X7, X8)
=P (X1)P (X2)P (X3|X1)P (X4|X2)P (X5|X2)
P (X6|X3, X4)P (X7|X6)P (X8|X5, X6)
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OtherdesiredpropertiesofPGMS
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•Modularity– Allowsustointegrateheterogeneousdata
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OtherdesiredpropertiesofPGMS
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• Priorknowledge– Bayesianlearning
• Capturesuncertaintyinamoreprincipledway– introducepriors
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Whatisagraphicalmodel?
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Multivariatestatistics+structure
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Whatisagraphicalmodel?
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• Informal: Itisasmartwaytospecify/compose/designexponentially-largeprobabilitydistributionswithoutpayinganexponentialcost,andatthesametimeendowthedistributionswithstructuredsemantics
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Whatisagraphicalmodel?
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•Moreformal: Itreferstoafamilyofdistributionsonasetofrandomvariablesthatarecompatiblewithalltheprobabilisticindependencepropositionsencodedbyagraphthatconnectsthesevariables
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TypesofPGMs
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•Directed: Bayesiannetworks•Directededgesgivecausalityrelationships
P (X1, X2, X3, X4, X5, X6, X7, X8)
=P (X1)P (X2)P (X3|X1)P (X4|X2)P (X5|X2)
P (X6|X3, X4)P (X7|X6)P (X8|X5, X6)
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TypesofPGMs
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•Undirected:Markovrandomfields•Undirectededgessimplygivecorrelationsbetweenvariables
P (X1, X2, X3, X4, X5, X6, X7, X8)
=
1
Zexp(E(X1) + E(X2) + E(X1, X3) + E(X2, X4) + E(X2, X5))
exp(E(X6, X3, X4) + E(X7, X6) + E(X8, X5, X6))
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BayesianNetworks
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•Structure:DAGs•Meaning:anodeisconditionallyindependentofeveryothernodeinthenetworkoutsideitsMarkovblanket
TheMarkovblanketofnode includesitsparents,childrenandtheotherparentsofallofitschildren.
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BayesianNetworks
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• Structure:DAGs
•Meaning:anodeisconditionallyindependentofeveryothernodeinthenetworkoutsideitsMarkovblanket
• Localconditionaldistributions(CPD)andtheDAGcompletelydeterminethejointdistribution.
• Edgesrepresentcausalityrelationships,andfacilitateagenerativeprocess
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MarkovRandomFields
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• Structure:undirectedgraph
•Meaning:anodeisconditionallyindependentofeveryothernodeinthenetworkgivenitsdirectneighbors
• Localcontingencyfunctions(potentials)andthecliquesinthegraphcompletelydeterminethejointdist.
• Edgesrepresentcorrelationsbetweenvariables,butnoexplicitwaytogeneratesamples
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Well-knownmodelsasPGMs
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•Densityestimation• Parametricandnon-parametricmethods
•Regression• Linear,conditionalmixture,non-parametric
•Classification• Generativeanddiscriminativeapproaches
•Clustering
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Morecomplexmodels
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•PartiallyobservedMarkovdecisionprocesses
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Morecomplexmodels
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• InformationExtraction
[OpenTag,Zhengetal.,KDD2018]
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Morecomplexmodels
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•Solidstatephysics
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ApplicationsofGraphicalModels
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• Machinelearning• Computationalstatistics• Computervisionandgraphics• NLP• Informationextraction• Roboticcontrol• Decisionmakingunderuncertainty• Computationalbiology• Medicaldiagnosis/prognosis• Financeandeconomics• Etc.
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WhyPGMs?
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•Languageforcommunication•Languageforcomputation•Languagefordevelopment
•Doesitremindyouofsomething?
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WhyPGMs?
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•Probabilitytheory: Formalframeworktocombineheterogeneouspartsandensureconsistency.•Graphstructure: Appealinginterfaceformodelinghighly-interactingsetsofvariables.Interpretabilityanddomainknowledge.•Generalization:ManyclassicalprobabilisticsystemsarespecialcasesofPGMs
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PGMsintheDeepLearningera
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•ProbabilisticModels: Goalistocapturethejointdistributionofinputvariables,outputvariables,latentvariables,parametersandhyper-parameters.Everythingisarandomvariables.•Deep(Learning)Models: Hierarchicalmodelstructurewheretheoutputofonemodelbecomestheinputofthenexthigherlevelmodel.Targetedtowardsfeaturelearning.
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PGMsintheDeepLearningera
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DeepLearning PGMsEmpiricalgoal: e.g.,Classification,featurelearning e.g.,transferlearning,latentvariable
inference
Structure: Graphical Graphical
Objective: Aggregatedfromlocalfunctions Aggregatedfromlocalfunctions
Vocabulary: Neuron, activation/gatefunction Variables,potentialfunctions
Algorithm: Singleinference algorithm,BP Manyalgorithms,major focusofopenresearch,approximateinference
Evaluation: Onend-performance On almosteveryintermediatequantity(calibratedprobabilities)
Implementation: ManytricksJ Quite standardized
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PGMsintheDeepLearningera
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•WhyProbabilisticModels?: Predictionsfromaprobabilisticmodelthatcapturesaprinciplednotionofuncertainty.Decisionmaking.•WhyDeep(Learning)Models: Featurelearning.Noassumptionsforcomplexdomainssuchasimagesandspeech.
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CombiningPGMsandDeepLearning
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•DeepBoltzmannMachines
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UsingPGMstogeneratetrainingdataforDL
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•Weaksupervision/Dataprogramming
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ClassOverview
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• FundamentalsofPGMs:• BayesianNetworksandMarkovRandomFields• Discrete,Continuous,andHybridmodels,exponentialfamily• Basicrepresentation,inference,andlearning• Focusonspecificnetworks:MultivariateGaussianModels,HiddenMarkovModels
• AdvancedTopics:• Approximateinference• Boundedtreewidth• SpectralmethodsforGraphicalmodels• Structurelearning• Relationalrepresentationlearningandconnectionstodeeplearning
• Applications