Parameter Learning of Logic Programs for Symbolic-Statistical Modeling

TL;DR

This paper introduces a unified logical framework and a new EM algorithm for statistical parameter learning in logic programs, enabling effective modeling of complex probabilistic systems like HMMs and Bayesian networks.

Contribution

It extends logic programming semantics to distribution semantics and proposes the graphical EM algorithm for efficient parameter learning in parameterized logic programs.

Findings

The graphical EM algorithm generalizes existing EM algorithms for various models.

It has comparable time complexity to traditional algorithms like Baum-Welch and Inside-Outside.

Experiments show it outperforms the Inside-Outside algorithm on PCFGs.

Abstract

We propose a logical/mathematical framework for statistical parameter learning of parameterized logic programs, i.e. definite clause programs containing probabilistic facts with a parameterized distribution. It extends the traditional least Herbrand model semantics in logic programming to distribution semantics, possible world semantics with a probability distribution which is unconditionally applicable to arbitrary logic programs including ones for HMMs, PCFGs and Bayesian networks. We also propose a new EM algorithm, the graphical EM algorithm, that runs for a class of parameterized logic programs representing sequential decision processes where each decision is exclusive and independent. It runs on a new data structure called support graphs describing the logical relationship between observations and their explanations, and learns parameters by computing inside and outside probability generalized for logic programs. The complexity analysis shows that when combined with OLDT search for all explanations for observations, the graphical EM algorithm, despite its generality, has the same time complexity as existing EM algorithms, i.e. the Baum-Welch algorithm for HMMs, the Inside-Outside algorithm for PCFGs, and the one for singly connected Bayesian networks that have been developed independently in each research field. Learning experiments with PCFGs using two corpora of moderate size indicate that the graphical EM algorithm can significantly outperform the Inside-Outside algorithm.