# A Differential Approach to Inference in Bayesian Networks

arXiv: https://arxiv.org/abs/1301.3847 · HTML: https://mlsanity.com/paper/1301.3847/

## Key facts

- Cited by 93 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2013.

## Built on by (most cited first, 30 of 93)

- [Training and Inference on Any-Order Autoregressive Models the Right Way](https://mlsanity.com/paper/2205.13554/) (2022)
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- [Safe Reinforcement Learning via Probabilistic Logic Shields](https://mlsanity.com/paper/2303.03226/) (2023)
- [Tractable Regularization of Probabilistic Circuits](https://mlsanity.com/paper/2106.02264/) (2021)
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- [Logically Consistent Language Models via Neuro-Symbolic Integration](https://mlsanity.com/paper/2409.13724/) (2024)
- [Neuro-Symbolic Entropy Regularization](https://mlsanity.com/paper/2201.11250/) (2022)
- [Probabilistic Generating Circuits](https://mlsanity.com/paper/2102.09768/) (2021)
- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Continuous Mixtures of Tractable Probabilistic Models](https://mlsanity.com/paper/2209.10584/) (2022)
- [Sparse Probabilistic Circuits via Pruning and Growing](https://mlsanity.com/paper/2211.12551/) (2022)
- [KLay: Accelerating Arithmetic Circuits for Neurosymbolic AI](https://mlsanity.com/paper/2410.11415/) (2024)
- [HyperSPNs: Compact and Expressive Probabilistic Circuits](https://mlsanity.com/paper/2112.00914/) (2021)
- [On the Relationship Between Monotone and Squared Probabilistic Circuits](https://mlsanity.com/paper/2408.00876/) (2024)
- [Causal Inference Using Tractable Circuits](https://mlsanity.com/paper/2202.02891/) (2022)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [Polynomial Semantics of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2402.09085/) (2024)
- [Solving Marginal MAP Exactly by Probabilistic Circuit Transformations](https://mlsanity.com/paper/2111.04833/) (2021)
- [Multi-objective Explanations of GNN Predictions](https://mlsanity.com/paper/2111.14651/) (2021)
- [Compositional Probabilistic and Causal Inference using Tractable Circuit Models](https://mlsanity.com/paper/2304.08278/) (2023)
- [A Compositional Atlas for Algebraic Circuits](https://mlsanity.com/paper/2412.05481/) (2024)
- [Restricted Boltzmann Machines as Models of Interacting Variables](https://mlsanity.com/paper/2103.15917/) (2021)
- [Tractable Boolean and Arithmetic Circuits](https://mlsanity.com/paper/2202.02942/) (2022)
- [COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuits](https://mlsanity.com/paper/2403.11348/) (2024)
- [Scaling Probabilistic Circuits via Monarch Matrices](https://mlsanity.com/paper/2506.12383/) (2025)
- [Fast and Expressive Multi-Byte Prediction with Probabilistic Circuits](https://mlsanity.com/paper/2511.11346/) (2025)
- [Leveraging Probabilistic Circuits for Nonparametric Multi-Output Regression](https://mlsanity.com/paper/2106.08687/) (2021)

## Abstract (arXiv)

We present a new approach for inference in Bayesian networks, which is mainly based on partial differentiation. According to this approach, one compiles a Bayesian network into a multivariate polynomial and then computes the partial derivatives of this polynomial with respect to each variable. We show that once such derivatives are made available, one can compute in constant-time answers to a large class of probabilistic queries, which are central to classical inference, parameter estimation, model validation and sensitivity analysis. We present a number of complexity results relating to the compilation of such polynomials and to the computation of their partial derivatives. We argue that the combined simplicity, comprehensiveness and computational complexity of the presented framework is unique among existing frameworks for inference in Bayesian networks.