Random Worlds and Maximum Entropy

TL;DR

This paper introduces the random-worlds method for computing degrees of belief from a knowledge base, showing its connection to maximum entropy in unary predicate cases and highlighting limitations in more complex scenarios.

Contribution

It establishes a formal link between the random-worlds approach and maximum entropy for unary predicates, revealing scope limitations and theoretical insights.

Findings

Degree of belief converges as domain size grows large.

Maximum entropy applies naturally in unary predicate cases.

Limitations arise in non-unary predicate scenarios.

Abstract

Given a knowledge base KB containing first-order and statistical facts, we consider a principled method, called the random-worlds method, for computing a degree of belief that some formula Phi holds given KB. If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or first-order models, with domain {1,...,N} that satisfy KB, and compute the fraction of them in which Phi is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying Phi and KB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger, there are many more worlds with higher entropy. Therefore, we can use a maximum-entropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are the limitations on its scope. Most importantly, the restriction to unary predicates seems necessary. Although the random-worlds method makes sense in general, the connection to maximum entropy seems to disappear in the non-unary case. These observations suggest unexpected limitations to the applicability of maximum-entropy methods.