Probabilistic Circuits for Cumulative Distribution Functions

TL;DR

This paper explores how probabilistic circuits can efficiently compute and transform between probability distribution functions and cumulative distribution functions for various types of variables, enhancing probabilistic inference capabilities.

Contribution

It demonstrates polynomial-time transformations between PMF and CDF in probabilistic circuits for binary, discrete, and continuous variables, extending their utility.

Findings

PMF and CDF are equivalent for binary variables in probabilistic circuits.

Transformations between PDFs and CDFs are efficient for discrete variables with circuit modifications.

Smooth, decomposable PCs for continuous variables can be converted between PDFs and CDFs by adjusting leaves.

Abstract

A probabilistic circuit (PC) succinctly expresses a function that represents a multivariate probability distribution and, given sufficient structural properties of the circuit, supports efficient probabilistic inference. Typically a PC computes the probability mass (or density) function (PMF or PDF) of the distribution. We consider PCs instead computing the cumulative distribution function (CDF). We show that for distributions over binary random variables these representations (PMF and CDF) are essentially equivalent, in the sense that one can be transformed to the other in polynomial time. We then show how a similar equivalence holds for distributions over finite discrete variables using a modification of the standard encoding with binary variables that aligns with the CDF semantics. Finally we show that for continuous variables, smooth, decomposable PCs computing PDFs and CDFs can be efficiently transformed to each other by modifying only the leaves of the circuit.