An asymptotic analysis of probabilistic logic programming, with implications for expressing projective families of distributions

TL;DR

This paper analyzes the asymptotic behavior of probabilistic logic programs, showing they are equivalent to acyclic programs with determinate clauses, and explores implications for expressing projective distributions in statistical relational AI.

Contribution

It proves that all probabilistic logic programs are asymptotically equivalent to acyclic, determinate clause programs, and introduces an abstract distribution semantics linking logic and probability.

Findings

Probabilistic logic programs are asymptotically equivalent to acyclic, determinate clause programs.

Programs inducing projective distributions are equivalent to a specific fragment of logic programs.

Introduces an abstract semantics bridging finite model theory and probabilistic logic programming.

Abstract

Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of statistical relational representations has come under scrutiny, and projectivity was investigated as the strongest form of domain-size dependence, in which query marginals are completely independent of the domain size. In this contribution we show that every probabilistic logic program under the distribution semantics is asymptotically equivalent to an acyclic probabilistic logic program consisting only of determinate clauses over probabilistic facts. We conclude that every probabilistic logic program inducing a projective family of distributions is in fact everywhere equivalent to a program from this fragment, and we investigate the consequences for the projective families of distributions expressible by probabilistic logic programs. To facilitate the application of classical results from finite model theory, we introduce the abstract distribution semantics, defined as an arbitrary logical theory over probabilistic facts. This bridges the gap to the distribution semantics underlying probabilistic logic programming. In this representation, determinate logic programs correspond to quantifier-free theories, making asymptotic quantifier elimination results available for the setting of probabilistic logic programming. This paper is under consideration for acceptance in TPLP.