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A Differential Approach to Inference in Bayesian Networks

arXiv 1301.3847 ↗First posted 16 Jan 2013cs.AIcited by 93

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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.

Key facts. Cited by 93 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Jan 2013.

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Citing papers per year: 2021: 13; 2022: 18; 2023: 11; 2024: 23; 2025: 14; 2026: 14.

93 papers.

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  1. 2022Training and Inference on Any-Order Autoregressive Models the Right Waycited by 64
  2. 2023Safe Reinforcement Learning via Probabilistic Logic Shieldscited by 41
  3. 2024R2-Guard: Robust Reasoning Enabled LLM Guardrail via Knowledge-Enhanced Logical Reasoningcited by 41
  4. 2021Tractable Regularization of Probabilistic Circuitscited by 39
  5. 2022Scaling Up Probabilistic Circuits by Latent Variable Distillationcited by 35
  6. 2024Logically Consistent Language Models via Neuro-Symbolic Integrationcited by 30
  7. 2022Neuro-Symbolic Entropy Regularizationcited by 27
  8. 2021Probabilistic Generating Circuitscited by 24
  9. 2024Scaling Tractable Probabilistic Circuits: A Systems Perspectivecited by 23
  10. 2021Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Modelscited by 22
  11. 2024What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?cited by 22
  12. 2024Sum of Squares Circuitscited by 21
38 more
  1. 2022Continuous Mixtures of Tractable Probabilistic Modelscited by 19
  2. 2022Sparse Probabilistic Circuits via Pruning and Growingcited by 17
  3. 2024KLay: Accelerating Arithmetic Circuits for Neurosymbolic AIcited by 16
  4. 2021HyperSPNs: Compact and Expressive Probabilistic Circuitscited by 14
  5. 2024On the Relationship Between Monotone and Squared Probabilistic Circuitscited by 14
  6. 2022Causal Inference Using Tractable Circuitscited by 12
  7. 2024Building Expressive and Tractable Probabilistic Generative Models: A Reviewcited by 12
  8. 2024Polynomial Semantics of Tractable Probabilistic Circuitscited by 12
  9. 2021Solving Marginal MAP Exactly by Probabilistic Circuit Transformationscited by 11
  10. 2021Multi-objective Explanations of GNN Predictionscited by 10
  11. 2023Compositional Probabilistic and Causal Inference using Tractable Circuit Modelscited by 10
  12. 2024A Compositional Atlas for Algebraic Circuitscited by 10
  13. 2021Restricted Boltzmann Machines as Models of Interacting Variablescited by 9
  14. 2022Tractable Boolean and Arithmetic Circuitscited by 9
  15. 2024COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuitscited by 9
  16. 2025Scaling Probabilistic Circuits via Monarch Matricescited by 9
  17. 2025Fast and Expressive Multi-Byte Prediction with Probabilistic Circuitscited by 9
  18. 2021Leveraging Probabilistic Circuits for Nonparametric Multi-Output Regressioncited by 8
  19. 2023Characteristic Circuitscited by 8
  20. 2022On the Complexity of Counterfactual Reasoningcited by 7
  21. 2023Bayesian Structure Scores for Probabilistic Circuitscited by 7
  22. 2023Tractable Probabilistic Graph Representation Learning with Graph-Induced Sum-Product Networkscited by 7
  23. 2024Probabilistic Neural Circuitscited by 7
  24. 2024Restructuring Tractable Probabilistic Circuitscited by 7
  25. 2021Provable Guarantees on the Robustness of Decision Rules to Causal Interventionscited by 6
  26. 2025How to Square Tensor Networks and Circuits Without Squaring Themcited by 5
  27. 2022You Only Derive Once (YODO): Automatic Differentiation for Efficient Sensitivity Analysis in Bayesian Networkscited by 4
  28. 2023Generative Marginalization Modelscited by 4
  29. 2024Probabilistic Inference in the Era of Tensor Networks and Differential Programmingcited by 4
  30. 2025Tractable Representation Learning with Probabilistic Circuitscited by 4
  31. 2022Certifying Fairness of Probabilistic Circuitscited by 3
  32. 2023Mixtures of All Treescited by 3
  33. 2024Identifying Causal Effects Under Functional Dependenciescited by 3
  34. 2024Causal Unit Selection using Tractable Arithmetic Circuitscited by 3
  35. 2024Sum-Product-Set Networks: Deep Tractable Models for Tree-Structured Graphscited by 3
  36. 2025Tractable Sharpness-Aware Learning of Probabilistic Circuitscited by 3
  37. 2025The DeepLog Neurosymbolic Machinecited by 3
  38. 2022Gradient-based Counterfactual Explanations using Tractable Probabilistic Modelscited by 2

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