First-Order Bayesian Network Specifications Capture the Complexity Class PP

TL;DR

This paper establishes a precise correspondence between the complexity class PP and the inferences in a specific class of Bayesian networks defined by first-order logic, linking computational complexity with probabilistic graphical models.

Contribution

It proves that PP can be characterized exactly by inferences in first-order Bayesian networks, providing a logical and probabilistic framework for understanding PP.

Findings

PP equals the set of languages encoded by valid inferences in first-order Bayesian networks.

First-order Bayesian networks can represent complex probabilistic inferences.

The result bridges computational complexity and probabilistic logic.

Abstract

The point of this note is to prove that a language is in the complexity class PP if and only if the strings of the language encode valid inferences in a Bayesian network defined using function-free first-order logic with equality.