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Probabilistic Logic Programming and Bayesian Networks

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Probabilistic Logic Programming and Bayesian Networks Probabilistic Logic Programming and Bayesian Networks Liem Ngo and Peter Haddawy Department of Electrical Engineering and Computer Science University of WisconsinMilwaukee Milwaukee WI fliem haddawygcsuwmedu Abstract We present a probabilistic logic programming framework that allows the repre sentation of conditional probabilities While conditional probabilities are the most commonly used metho d for representing uncertainty in probabilistic exp ert systems they have b een largely neglected bywork in quantitative logic programming We de ne a xp oint theory declarativesemantics and pro of pro cedure for the new class of probabilistic logic programs Compared to other approaches to quantitative logic programming weprovide a true probabilistic framework with p otential applications in probabilistic exp ert systems and decision supp ort systems We also discuss the relation ship b etween such programs and Bayesian networks thus moving toward a unication of two ma jor approaches to automated reasoning To appear in Proceedings of the Asian Computing Science Conference Pathumthani Thailand December This work was partially supp orted by NSF grant IRI Intro duction Reasoning under uncertainty is a topic of great imp ortance to many areas of Computer Sci ence Of all approaches to reasoning under uncertainty probability theory has the strongest theoretical foundations In the quest to extend the framwork of logic programming to represent and reason with uncertain knowledge there havebeenseveral attempts to add numeric representations of uncertainty to logic programming languages Of these attempts the only one to use probabilityisthework of Ng and Sub rahmanian In their framework a probabilistic logic program is an annotated Horn program Atypical example clause in a probabilistic logic program taken from is pathX Y aX Y whichsays that if the probabilitythata typ e A connection is used lies in the interval then the reliability of the path is b etween and As this example illustrates their framework do es not employ conditional probability which is the most common waytoquantify degrees of inuence in probabilistic reasoning and probabilistic exp ert systems In the authors allow clauses to b e interpreted as ts but they consider only the consistency of such programs conditional probability statemen and do not provide a query answering pro cedure Bayesian networks have b ecome the most p opular metho d for representing and rea soning with probabilistic information An extended form of Bayesian networks inuence diagrams are widely used in decision analysis The strengths of causal relationships in Bayesian networks and inuence diagrams are sp ecied with conditional probabilities A prominent feature of Bayesian networks is that they allow computation of p osterior proba bilities and p erformance of systematic sensitivity analysis which is imp ortant when the exact probabilityvalues are hard to obtain Bayesian networks are used as the main representation and reasoning device in probabilistic diagnostic systems and exp ert systems Bayesian networks were originally presented as static graphical mo dels for a problem domain the relevant random variables are identied a Bayesian network representing the relationships b etween the random variables is sketched and probabilityvalues are assessed Inference is then p erformed using the entire domain mo del even if only a p ortion is relevantto agiven inference problem Recently the approachknown as knowledgebased mo del construc tion has attempted to address this limitation by representing probabilistic information in wledge base using schematic variables and indexing schemes and constructing a network a kno mo del tailored to each sp ecic problem The constructed network is a subset of the domain mo del represented by the collection of sentences in the knowledge base Approaches to this area of researchhave either fo cused on practical mo del construction algorithms neglecting formal asp ects of the problem or fo cused on formal asp ects of the knowledge base rep resentation language without presenting practical algorithms for constructing networks In we prop ose b oth a theoretical framework and a pro cedure for constructing Bayesian networks from a set of conditional probabilistic sentences The purp ose of this pap er is twofold First we prop ose an extension of logic program ming which allows the representation of conditional probabilities and hence can b e used to write probabilistic exp ert systems Second weinvestigate the relationship b etween prob abilistic logic programs and Bayesian networks While Poole shows how to represent a discrete Bayesian network in his Probabilistic Horn Ab duction framework in this pap er we address b oth sides of this relationship First weshowhowBayesian networks can b e represented easily and intuitively by our probabilistic logic programs Second we presenta metho d for answering queries on the probabilistic logic programs by constructing Bayesian networks and then prop ogating probabilities on the networks Weprovide a declarative se mantics for probabilistic logic programs and prove that the constructed Bayesian networks faithfully reect the declarative semantics Syntax C Throughout this pap er we use Pr and sometimes Pr Pr to denote a probability distribu P P tion AB with p ossible subscripts to denote atoms names with leading capital characters to denote domain variables names with leading small characters to denote constants and pq with p ossible subscripts to denote predicates We use a rst order language con taining innitely manyvariable symb ols and nitely many constant function and predicate symb ols WeuseHB to denote the Herbrand base of the language which can b e innite For convenience we use comma instead of logical AND and semicolons to sep erate the sentences in a list of sentences Each predicate represents a class of similar random variables In the probability mo dels we consider eac h random variable can takevalues from a nite set and in each p ossible realization of the world that variable can have one and only one value For example the variable neighborhood of a p erson X can havevalue bad average good and in each p ossible realization of the world one and only one of these three values can b e true the others must b e false We capture this prop ertyby requiring that each predicate have at least one attribute representing the value of the corresp onding random variable By convention we take this to b e the last attribute For example the variable neighborhood of a p erson X can b e represented byatwop osition predicate neig hbor hoodX V the rst p osition indicates the p erson and the second indicates the typ e of that p ersons neighb orho o d bad average or go o d We asso ciate with each predicate a value integrity constraint statement Denition The value integrity constraint statement associated with an mary pred icate p consists of the fol lowing rst order sentences pX X V V v m V v pX X v pX X v i j i j n where n m i m j nm are two integers v v are dierent constants cal ledthevalueconstants n denoting the possible values of the random variables corresponding to p X X are m e is universal ly quantied over the entire sentence dierent variable names and each sentenc For convenience we use EXCLUSIV Ep v v to denote the above set of sentences n We use as the identity relation on HB and always assume our theories include Clarks Equality Theory We denote by VALp the set fv v g If A is an atom of predicate n pwe also use VALA as equivalentto VALp If A is the ground atom pt t v m then valA denotes the value v and obj A denotes the random variable corresp onding to p t t m We require sucha value integrity constraint for each predicate The set of all the integrity constraints is denoted byIC Example The value integrity constraint for the predicate neig hbor hood is EX CLUSI V Eneig hbor hood bad av erag e g ood f neig hbor hoodX bad neig hbor hoodX av er ag e neig hbor hoodX bad neig hbor hoodX g ood neig hbor hoodX av er ag e neig hbor hoodX g ood neig hbor hoodX V V bad V av er ag e V g oodg For a person say named John neig hbor hoodj ohn g ood means the random variable neigh borhood of John indicated in the language by obj neig hbor hoodj ohn g ood isgo o d indi cated in the language by valneig hbor hoodj ohn g ood goodInanypossible world one and only one of the fol lowing atoms is true neig hbor hoodj ohn bad neig hbor hoodjohn averageorneig hbor hoodj ohn g ood VALneig hbor hoodor VALneig hbor hoodjohn bad is the set fbad av er ag e g oodg Wehavetwo kinds of constants The value constants are declared byEXCLUSIVE clauses and used as the last arguments of predicates The nonvalue constants are used for the other predicate arguments Denition LetAbe the ground atom pt t We dene ExtA the extension of m pg A to be the set fpt t vjv VAL m Example In the burglary example Extneig hbor hoodj ohn bad fneig hbor hoodj ohn bad neig hbor hoodj ohn av er ag e neig hbor hoodj ohn g ood g Let A b e an atom We dene groundA to b e the set of all ground instances of AA set of ground atoms fA j i ng is called coherent if there do not exist any A and A i j j and valA valA such that j j and obj A obj A j j j j Denition A probabilistic sentence has the form PrA jA A where n n andA are atoms The sentencecan have free variables and each free vari i able is universal ly quantied over its entirescope The meaning of such a sentenceisIf PrB
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