# Probabilistic Generating Circuits

arXiv: https://arxiv.org/abs/2102.09768 · HTML: https://mlsanity.com/paper/2102.09768/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Strudel: Learning Structured-Decomposable Probabilistic Circuits](https://mlsanity.com/paper/2007.09331/) (2020)
- [Learning the Parameters of Determinantal Point Process Kernels](https://mlsanity.com/paper/1402.4862/) (2014)
- [Expectation-Maximization for Learning Determinantal Point Processes](https://mlsanity.com/paper/1411.1088/) (2014)
- [On the Expressive Efficiency of Sum Product Networks](https://mlsanity.com/paper/1411.7717/) (2014)
- [Learning Nonsymmetric Determinantal Point Processes](https://mlsanity.com/paper/1905.12962/) (2019)
- [Fixed-point algorithms for learning determinantal point processes](https://mlsanity.com/paper/1508.00792/) (2015)
- [Smoothing Structured Decomposable Circuits](https://mlsanity.com/paper/1906.00311/) (2019)
- [Large-Margin Determinantal Point Processes](https://mlsanity.com/paper/1411.1537/) (2014)
- [Fast Mixing Markov Chains for Strongly Rayleigh Measures, DPPs, and Constrained Sampling](https://mlsanity.com/paper/1608.01008/) (2016)
- [On the Relationship Between Probabilistic Circuits and Determinantal Point Processes](https://mlsanity.com/paper/2006.15233/) (2020)
- [AND/OR Multi-Valued Decision Diagrams (AOMDDs) for Graphical Models](https://mlsanity.com/paper/1401.3448/) (2014)
- [Kronecker Determinantal Point Processes](https://mlsanity.com/paper/1605.08374/) (2016)
- [Flexible Modeling of Diversity with Strongly Log-Concave Distributions](https://mlsanity.com/paper/1906.05413/) (2019)
- [The Libra Toolkit for Probabilistic Models](https://mlsanity.com/paper/1504.00110/) (2015)

## Built on by (most cited first, 25 of 25)

- [Tractable Control for Autoregressive Language Generation](https://mlsanity.com/paper/2304.07438/) (2023)
- [Adversarial random forests for density estimation and generative modeling](https://mlsanity.com/paper/2205.09435/) (2022)
- [Subtractive Mixture Models via Squaring: Representation and Learning](https://mlsanity.com/paper/2310.00724/) (2023)
- [Sparse Probabilistic Circuits via Pruning and Growing](https://mlsanity.com/paper/2211.12551/) (2022)
- [HyperSPNs: Compact and Expressive Probabilistic Circuits](https://mlsanity.com/paper/2112.00914/) (2021)
- [Lifted Inference with Linear Order Axiom](https://mlsanity.com/paper/2211.01164/) (2022)
- [Polynomial Semantics of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2402.09085/) (2024)
- [Characteristic Circuits](https://mlsanity.com/paper/2312.07790/) (2023)
- [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)
- [Hardness of Maximum Likelihood Learning of DPPs](https://mlsanity.com/paper/2205.12377/) (2022)
- [Exact Bayesian Inference on Discrete Models via Probability Generating Functions: A Probabilistic Programming Approach](https://mlsanity.com/paper/2305.17058/) (2023)
- [Probabilistic Generating Circuits -- Demystified](https://mlsanity.com/paper/2404.02912/) (2024)
- [Rethinking Probabilistic Circuit Parameter Learning](https://mlsanity.com/paper/2505.19982/) (2025)
- [Mixtures of All Trees](https://mlsanity.com/paper/2302.14202/) (2023)
- [On the Expressive Power of Tree-Structured Probabilistic Circuits](https://mlsanity.com/paper/2410.05465/) (2024)
- [Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity](https://mlsanity.com/paper/2605.07097/) (2026)
- [On the Hardness of Approximating Distributions with Tractable Probabilistic Models](https://mlsanity.com/paper/2506.01281/) (2025)
- [The Limits of Tractable Marginalization](https://mlsanity.com/paper/2506.12020/) (2025)
- [Sum-Product-Attention Networks: Leveraging Self-Attention in Probabilistic Circuits](https://mlsanity.com/paper/2109.06587/) (2021)
- [Probabilistic Circuits for Cumulative Distribution Functions](https://mlsanity.com/paper/2408.04229/) (2024)
- [Tractable Probabilistic Models for Investment Planning](https://mlsanity.com/paper/2511.13888/) (2025)
- [Learning Tractable Distributions Of Language Model Continuations](https://mlsanity.com/paper/2511.16054/) (2025)
- [Variance Computation for Weighted Model Counting with Knowledge Compilation Approach](https://mlsanity.com/paper/2601.03523/) (2026)
- [Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels](https://mlsanity.com/paper/2610.12894/) (2026)

## Abstract (arXiv)

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.