Polynomial Semantics of Tractable Probabilistic Circuits

TL;DR

This paper demonstrates the polynomial equivalence of various probabilistic circuit semantics for binary variables, establishing their tractability for marginal inference, and explores the complexity extension to categorical variables.

Contribution

It proves the polynomial equivalence of different polynomial semantics in probabilistic circuits for binary variables and extends the framework to categorical variables, analyzing inference complexity.

Findings

All considered probabilistic circuit models are polynomially transformable.

These models are tractable for marginal inference over the same distribution class.

Inference becomes #P-hard when extending to categorical variables.

Abstract

Probabilistic circuits compute multilinear polynomials that represent multivariate probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in the literature (e.g., network polynomials, likelihood polynomials, generating functions, and Fourier transforms). The relationships between circuit representations of these polynomial encodings of distributions is largely unknown. In this paper, we prove that for distributions over binary variables, each of these probabilistic circuit models is equivalent in the sense that any circuit for one of them can be transformed into a circuit for any of the others with only a polynomial increase in size. They are therefore all tractable for marginal inference on the same class of distributions. Finally, we explore the natural extension of one such polynomial semantics, called probabilistic generating circuits, to categorical random variables, and establish that inference becomes #P-hard.