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Probabilistic Circuits for Cumulative Distribution Functions

Oliver Broadrick, William Cao, Benjie Wang, Martin Trapp, Guy Van den Broeck

TL;DR

This work investigates using the cumulative distribution function (CDF) as a semantics for probabilistic circuits (PCs), examining equivalence with the traditional PMF/PDF semantics across three regimes. In the binary case, the CDF polynomial coincides with the probability generating function (PGF), enabling polynomial-time transformations between PMF and CDF circuits. For finite discrete variables, a Less-Than Encoding LT_k preserves the required order so that the finite-discrete case reduces to the binary case and inherits the same transformations. In the continuous setting, smooth and decomposable PCs permit leaf-level modifications to transform between PDFs and CDFs, with two-way transformations supported under the stated structural assumptions. Together, these results establish practical, structurally light conditions under which CDF-based PCs are as expressive and tractable as PMF/PDF-based PCs, broadening the applicability of tractable probabilistic reasoning with PCs.

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.

Probabilistic Circuits for Cumulative Distribution Functions

TL;DR

This work investigates using the cumulative distribution function (CDF) as a semantics for probabilistic circuits (PCs), examining equivalence with the traditional PMF/PDF semantics across three regimes. In the binary case, the CDF polynomial coincides with the probability generating function (PGF), enabling polynomial-time transformations between PMF and CDF circuits. For finite discrete variables, a Less-Than Encoding LT_k preserves the required order so that the finite-discrete case reduces to the binary case and inherits the same transformations. In the continuous setting, smooth and decomposable PCs permit leaf-level modifications to transform between PDFs and CDFs, with two-way transformations supported under the stated structural assumptions. Together, these results establish practical, structurally light conditions under which CDF-based PCs are as expressive and tractable as PMF/PDF-based PCs, broadening the applicability of tractable probabilistic reasoning with PCs.

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.
Paper Structure (8 sections, 5 theorems, 26 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on circuits
  3. Cumulative Distribution Functions
  4. Binary Variables
  5. Finite Discrete Variables
  6. Continuous Variables
  7. Conclusion
  8. Proofs

Key Result

Proposition 1

Fix a probability distribution over binary random variables $X_1,\ldots,X_n$ taking values $\{0,1\}\subset \mathbb{R}$ with CDF polynomial $c$ and PGF $g$. Then $c=g$.

Figures (1)

  • Figure 1: A probability distribution over two binary random variables. The PMF and CDF functions are specified in the table, and the corresponding PMF and CDF polynomials $p$ and $c$ are given below.

Theorems & Definitions (11)

  • Definition 1
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 2
  • Definition 3: Smoothness
  • Definition 4: Decomposable
  • Proposition 2
  • Proposition 3
  • Lemma 1: BroadrickUAI24
  • ...and 1 more