Clausal Representations of Markov Networks and Related Probabilistic Models

Connections between Muggleton's stochastic logic programs (SLPs) and (undirected) Bayesian nets are explored. Logical variables map to random variables, predicates map to joint distributions and Markov net structure maps to clausal structure. Naive methods for answering proba-bilistic queries using SLPs are given, as well as some suggestions for non-naive approaches. Multi-clause SLPs deene mixture models, and these are used to express conditional conditional independence. The use of SLPs to implement probabilistic rst-order reasoning is brieey discussed. 1 Stochastic logic programs A stochastic logic program (SLP) 4] is a deenite logic program where some clauses are annotated with a non-negative number so that a probability distribution is associated with a set of distribution-deening predicates in the logic program. It is required that the computed answer substitutions for any unit goal where the predicate is distribution-deening be ground. Requiring all clauses to be range-restricted achieves this: in a range-restricted clause every variable that appears in the head must appear in the body. Let us call these annotations clause labels. We can use these clause labels to associate a number with any SLD-refutation of a goal G: it is simply the product of all the clause labels of the clauses used in that refutation. Let us call these refutation potentials, and denote the refutation potential of a refutation R by (R). If there are no clause labels associated with a refutation R 0 , we set (R 0) = 0. Consider all the refutations of a goal G, we deene the goal potential, (G) of G as follows (G) = X R is a refutation of G (R) (1) where (G) is undeened if the RHS is undeened (innnite). Clause labels deene an SLP ii (G) is deened for any goal. Note that goals not in the language of the program simply have a potential of zero. We now conjecture a suucient condition for clause labels to deene an SLP. Conjecture 1. If, for any predicate in an annotated logic program, the sum of the clause labels of the clauses that deene that predicate is at most one, then the annotated logic program is an SLP.