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CMU CS 10708 - lecture21

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1Probabilistic Graphical Models10-708Models with HigherModels with Higher--Level Level Structures: logic + probabilitiesStructures: logic + probabilitiesEric Xing Eric Xing Lecture 21, Nov 28, 2005Reading: Getoor et al 2001, Milch et al. 2005Limitations of GMz Applications are pushing the representation and modeling limits of GM …z Open domains with both structural and attribute uncertainty!Number uncertaintyRelational uncertaintyRecursive relationsRecursive relationsIdentity uncertaintyExistence uncertaintyAttribute uncertaintyAggregate functions2Propositional Logicz Ontological commitment: the world consists of propositions, or facts, or atomic events, which are either true or falsez e.g., Paper_X_HighPaperRatingz Set of 2npossible worlds – one for each truth assignment to the n propositionsz Propositional logic allows us to compactly represent restrictions on possible worlds: z If Auther_A_HighPublicationRating then Paper_X_HighPaperRatingz Means that we have eliminated the possible worlds where Auther_A_HighPublicationRating is true but Paper_X_HighPaperRating is false.Propositional Uncertaintyz To model uncertainty we would like to represent a probability distribution over all possible worlds.z To represent the full joint distribution we would need 2n-1 parameters (infeasible)z Insight: the value of most propositions isn't affected by the value of most other propositions!z More formally, some propositions are conditionally independent of each other given the value of other propositions3Bayesian Networksz A BN uses a directed acyclic graph to encode these independence assumptionsz This model encodes the assumption that each variable is independent of its non-descendents given its parents z The full joint over these five binary variables would need 25-1=31 parameters, but this factored representation only needs 10!AuthorInstitutionPaperRatingAuthorRatingJournalRatingPaperCited0.01P(AI=Stanford)0.01low0.5highP(PC=true | PR)PR0.001other0.1Stanf.P(AR=high | AI)AI0.3P(JR=high)otherotherStanf.Stanf.AI0.1low0.6high0.01low0.2highP(PR=high | AI, JR)JRPlates and beyondz Graphical model applies to any paper Æ already “universally quantified”z a Plate stands for N IID replicates of the enclosed model (Buntine 1994)z Can we reason across objects?z e.g., the rating of a paper authored by F. Crick given the ratings of some papers authored by J. WatsonAuthorInstitutionPaperRatingAuthorRatingJournalRatingPaperCitedNN4Shortcomings of Bayes Net z BNs lack the concept of an object z Cannot represent general rules about the relations between multiple similar objectsz For example, if we wanted to represent the probabilities over multiple papers, authors, and journals:z We would need an explicit random variable for each paper/author/journalz The distributions would be separate, so knowledge about one wouldn't impart any knowledge about the othersz BNs assume domain closure, unique name, and relational invariancez Can not represent open possible world with unknown number of objectsz Can not accommodate objects possibly with multiple namesz Can not succinctly represent uncertainty in data associationz …Statistical Relational Learningz In general, SRL combines logic and probabilitiesz Historically, there are two general threads of research1. Frame-based Probabilistic Modelsz Probabilistic Relational Models (PRMs), z Probabilistic Entity Relation Models (PERs), z Object Oriented Bayesian Networks (OOBNs)This thread takes graphical models or hierarchical Bayesian models and adds in some form of relational/logical representation2. First Order Probabilistic Logic (FOPL)z BLOGsz Relational Markov Logic (RML)This thread takes a logical representation (first-order logic, horn clauses, etc) and adds in some form of probabilities5Probabilistic Relational Models (PRMs)z Combine advantages of relational logic & Bayesian networks: z natural domain modeling: objects, properties, relations;z generalization over a variety of situations;z compact, natural probability models.z Integrate uncertainty with relational model:z properties of domain entities can depend on properties of related entities;z uncertainty over relational structure of domain.Motivation: Discovering Patterns in Structured DataPatientTreatmentStrainContact6From relational database to PRMDatabasePatientStrainContactRelational SchemaPatientContactStrain• Parameter estimation• Structure selectionStrainUniqueInfectivityInfected withInteracted withz Describes the types of objects and relations in the databaseClassesClassesRelationshipsRelationshipsContactClose-ContactSkin-TestAgePatientHomelessHIV-ResultEthnicityDisease-SiteAttributesAttributesContact-TypeRelational Schema7⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛Cont.Contactor.HIVCont.Close-ContactCont.Transmitted | PClose-ContactTransmittedContact-TypeDisease SiteStrainUniqueInfectivityPatientHomelessHIV-ResultPOBContactAge4.06.03.07.02.08.01.09.0,,,,,fftfftttP(T | H, C)CHProbabilistic Relational Model{})()),(())(( ,xxxxxcontactcloseceAcquaintanresultHIVdTransmitteParentsContact−−=⇒∈∀90.)(,))(()( ,=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=−=−=⇒∈∀truextruextruexxxcontactcloseceAcquaintanresultHIVdTransmittePContactSimple functionSimple functionComplex functionComplex functionz Complex functions specifies complex relations among objectsz Fixed relational skeleton σz set of objects in each classz relations between themz Uncertainty over assignment of values to attributes (AU)z PRM defines distribution over instantiations of attributesStrains1Patientp2Patientp1Contactc3Contactc2Contactc1Strains2Patientp3Relational Skeleton8P1.Disease SiteP1.HomelessP1.HIV-ResultP1.POBC1.Close-ContactC1.TransmittedC1.Contact-TypeC1.AgeC2.Close-ContactC2.TransmittedC2.Contact-Typetruefalsetrue4.06.03.07.02.08.01.09.0,,,,,ttfttfffP(T | H, C)CH4.06.03.07.02.08.01.09.0,,,,,ttfttfffP(T | H, C)CHC2.AgeA Portion of the BNz A PRM w/ AU and fixed, valid relations is equivalent to an unrolled BNsum, min, max, avg, mode, countDisease SitePatientHomelessHIV-ResultPOBAgeClose-ContactTransmittedContact-TypeContactAge..PatientJane DoePOB USHomeless noHIV-Result negativeAge ???Disease Site pulmonaryA.Contact#5077Contact-TypecoworkerClose-Contact no Agemiddle-agedTransmitted falseContact#5076Contact-TypespouseClose-Contact yes Agemiddle-agedTransmitted trueContact#5075Contact-TypefriendClose-Contact no Agemiddle-agedTransmitted falsemode6.03.01.02.06.02.02.04.04.0omyomymPRM: Aggregate


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CMU CS 10708 - lecture21

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