# Partition Function Estimation: A Quantitative Study

arXiv: https://arxiv.org/abs/2105.11132 · HTML: https://mlsanity.com/paper/2105.11132/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 108 references that are in the map.
- First posted on arXiv 24 May 2021.

## Builds on (most cited first, 19 of 19)

- [Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks](https://mlsanity.com/paper/1301.3853/) (2013)
- [Belief Propagation Neural Networks](https://mlsanity.com/paper/2007.00295/) (2020)
- [Amortized Inference Regularization](https://mlsanity.com/paper/1805.08913/) (2018)
- [The Model Counting Competition 2020](https://mlsanity.com/paper/2012.01323/) (2020)
- [Meta-Amortized Variational Inference and Learning](https://mlsanity.com/paper/1902.01950/) (2019)
- [Bucket Elimination: A Unifying Framework for Several Probabilistic Inference](https://mlsanity.com/paper/1302.3572/) (2013)
- [Variational Particle Approximations](https://mlsanity.com/paper/1402.5715/) (2014)
- [Taming the Curse of Dimensionality: Discrete Integration by Hashing and Optimization](https://mlsanity.com/paper/1302.6677/) (2013)
- [Sparse Hashing for Scalable Approximate Model Counting: Theory and Practice](https://mlsanity.com/paper/2004.14692/) (2020)
- [Differentiable Antithetic Sampling for Variance Reduction in Stochastic Variational Inference](https://mlsanity.com/paper/1810.02555/) (2018)
- [Simulation Approaches to General Probabilistic Inference on Belief Networks](https://mlsanity.com/paper/1304.1526/) (2013)
- [A Constraint Propagation Approach to Probabilistic Reasoning](https://mlsanity.com/paper/1304.3422/) (2013)
- [Bounded Conditioning: Flexible Inference for Decisions under Scarce Resources](https://mlsanity.com/paper/1304.1512/) (2013)
- [Optimization With Parity Constraints: From Binary Codes to Discrete Integration](https://mlsanity.com/paper/1309.6827/) (2013)
- [Efficient semidefinite-programming-based inference for binary and multi-class MRFs](https://mlsanity.com/paper/2012.02661/) (2020)
- [Approximate Inference via Weighted Rademacher Complexity](https://mlsanity.com/paper/1801.09028/) (2018)
- [Training Variational Autoencoders with Buffered Stochastic Variational Inference](https://mlsanity.com/paper/1902.10294/) (2019)
- [Intervention in undirected Ising graphs and the partition function](https://mlsanity.com/paper/1905.11502/) (2019)
- [Taming Discrete Integration via the Boon of Dimensionality](https://mlsanity.com/paper/2010.10724/) (2020)

## Built on by (most cited first, 8 of 8)

- [Causal Inference Using Tractable Circuits](https://mlsanity.com/paper/2202.02891/) (2022)
- [Tractable Boolean and Arithmetic Circuits](https://mlsanity.com/paper/2202.02942/) (2022)
- [Projected Model Counting: Beyond Independent Support](https://mlsanity.com/paper/2110.09171/) (2021)
- [Tractable Bounding of Counterfactual Queries by Knowledge Compilation](https://mlsanity.com/paper/2310.03352/) (2023)
- [IBIA: Bayesian Inference via Incremental Build-Infer-Approximate operations on Clique Trees](https://mlsanity.com/paper/2202.12003/) (2022)
- [IBIA: An Incremental Build-Infer-Approximate Framework for Approximate Inference of Partition Function](https://mlsanity.com/paper/2304.06366/) (2023)
- [An ASP-Based Framework for MUSes](https://mlsanity.com/paper/2507.03929/) (2025)
- [Answer Set Counting and its Applications](https://mlsanity.com/paper/2502.09231/) (2025)

## Abstract (arXiv)

Probabilistic graphical models have emerged as a powerful modeling tool for several real-world scenarios where one needs to reason under uncertainty. A graphical model's partition function is a central quantity of interest, and its computation is key to several probabilistic reasoning tasks. Given the #P-hardness of computing the partition function, several techniques have been proposed over the years with varying guarantees on the quality of estimates and their runtime behavior. This paper seeks to present a survey of 18 techniques and a rigorous empirical study of their behavior across an extensive set of benchmarks. Our empirical study draws up a surprising observation: exact techniques are as efficient as the approximate ones, and therefore, we conclude with an optimistic view of opportunities for the design of approximate techniques with enhanced scalability. Motivated by the observation of an order of magnitude difference between the Virtual Best Solver and the best performing tool, we envision an exciting line of research focused on the development of portfolio solvers.