# Probabilistic Generating Circuits -- Demystified

arXiv: https://arxiv.org/abs/2404.02912 · HTML: https://mlsanity.com/paper/2404.02912/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 4 Mar 2024.

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

- [Combating Label Noise in Deep Learning Using Abstention](https://mlsanity.com/paper/1905.10964/) (2019)
- [PAC-Bayes Un-Expected Bernstein Inequality](https://mlsanity.com/paper/1905.13367/) (2019)
- [Probabilistic Generating Circuits](https://mlsanity.com/paper/2102.09768/) (2021)
- [On the Expressive Efficiency of Sum Product Networks](https://mlsanity.com/paper/1411.7717/) (2014)
- [Smoothing Structured Decomposable Circuits](https://mlsanity.com/paper/1906.00311/) (2019)
- [Polynomial Semantics of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2402.09085/) (2024)
- [Split-kl and PAC-Bayes-split-kl Inequalities for Ternary Random Variables](https://mlsanity.com/paper/2206.00706/) (2022)
- [On the Relationship Between Probabilistic Circuits and Determinantal Point Processes](https://mlsanity.com/paper/2006.15233/) (2020)

## Built on by (most cited first, 4 of 4)

- [Restructuring Tractable Probabilistic Circuits](https://mlsanity.com/paper/2411.12256/) (2024)
- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [The Limits of Tractable Marginalization](https://mlsanity.com/paper/2506.12020/) (2025)
- [A Quantum Information Theoretic Approach to Tractable Probabilistic Models](https://mlsanity.com/paper/2506.01824/) (2025)

## Abstract (arXiv)

Zhang et al. (ICML 2021, PLMR 139, pp. 12447-1245) introduced probabilistic generating circuits (PGCs) as a probabilistic model to unify probabilistic circuits (PCs) and determinantal point processes (DPPs). At a first glance, PGCs store a distribution in a very different way, they compute the probability generating polynomial instead of the probability mass function and it seems that this is the main reason why PGCs are more powerful than PCs or DPPs. However, PGCs also allow for negative weights, whereas classical PCs assume that all weights are nonnegative. One of the main insights of our paper is that the negative weights are responsible for the power of PGCs and not the different representation. PGCs are PCs in disguise, in particular, we show how to transform any PGC into a PC with negative weights with only polynomial blowup. PGCs were defined by Zhang et al. only for binary random variables. As our second main result, we show that there is a good reason for this: we prove that PGCs for categorial variables with larger image size do not support tractable marginalization unless NP = P. On the other hand, we show that we can model categorial variables with larger image size as PC with negative weights computing set-multilinear polynomials. These allow for tractable marginalization. In this sense, PCs with negative weights strictly subsume PGCs.