# Probabilistic Circuits for Cumulative Distribution Functions

arXiv: https://arxiv.org/abs/2408.04229 · HTML: https://mlsanity.com/paper/2408.04229/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 8 Aug 2024.

## Builds on (most cited first, 8 of 8)

- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Probabilistic Generating Circuits](https://mlsanity.com/paper/2102.09768/) (2021)
- [Deep Archimedean Copulas](https://mlsanity.com/paper/2012.03137/) (2020)
- [Polynomial Semantics of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2402.09085/) (2024)
- [Scaling Integer Arithmetic in Probabilistic Programs](https://mlsanity.com/paper/2307.13837/) (2023)
- [Unsupervised tree boosting for learning probability distributions](https://mlsanity.com/paper/2101.11083/) (2021)
- [Bit Blasting Probabilistic Programs](https://mlsanity.com/paper/2312.05706/) (2023)

## Built on by (most cited first, 1 of 1)

- [The Limits of Tractable Marginalization](https://mlsanity.com/paper/2506.12020/) (2025)

## Abstract (arXiv)

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.