Conditional probability logic, lifted Bayesian networks, and almost sure quantifier elimination

Abstract

We introduce a formal logical language, called conditional probability logic (CPL), which extends first-order logic and which can express probabilities, conditional probabilities and which can compare conditional probabilities. Intuitively speaking, although formal details are different, CPL can express the same kind of statements as some languages which have been considered in the artificial intelligence community. We also consider a way of making precise the notion of lifted Bayesian network, where this notion is a type of (lifted) probabilistic graphical model used in machine learning, data mining and artificial intelligence. A lifted Bayesian network (in the sense defined here) determines, in a natural way, a probability distribution on the set of all structures (in the sense of first-order logic) with a common finite domain D. Our main result (Theorem 3.14) is that for every “noncritical” CPL-formula φ(x¯) there is a quantifier-free formula φ⁎(x¯) which is “almost surely” equivalent to φ(x¯) as the cardinality of D tends towards infinity. This is relevant for the problem of making probabilistic inferences on large domains D, because (a) the problem of evaluating, by “brute force”, the probability of φ(x¯) being true for some sequence d¯ of elements from D has, in general, (highly) exponential time complexity in the cardinality of D, and (b) the corresponding probability for the quantifier-free φ⁎(x¯) depends only on the lifted Bayesian network and not on D. Some conclusions regarding the computational complexity of finding φ⁎ are given in Remark 3.17. The main result has two corollaries, one of which is a convergence law (and zero-one law) for noncritial CPL-formulas.