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A Conceptual Introduction to Hamiltonian Monte Carlo

arXiv 1701.02434 ↗First posted 10 Jan 2017stat.MEcited by 207

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Abstract (arXiv)

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

Key facts. Cited by 207 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Jan 2017.

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Citing papers per year: 2021: 32; 2022: 40; 2023: 29; 2024: 46; 2025: 29; 2026: 31.

207 papers.

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  1. 2022Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Modelscited by 50
  2. 2021Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networkscited by 48
  3. 2021Adversarial Robustness with Semi-Infinite Constrained Learningcited by 39
  4. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
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  6. 2021Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priorscited by 29
  7. 2021Pathfinder: Parallel quasi-Newton variational inferencecited by 26
  8. 2022Langevin Diffusion Variational Inferencecited by 26
  9. 2022Markov Chain Monte Carlo-Based Machine Unlearning: Unlearning What Needs to be Forgottencited by 25
  10. 2021A Survey of Monte Carlo Methods for Parameter Estimationcited by 23
  11. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  12. 2022Missing Data Imputation and Acquisition with Deep Hierarchical Models and Hamiltonian Monte Carlocited by 19
38 more
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  4. 2024FlowPG: Action-constrained Policy Gradient with Normalizing Flowscited by 16
  5. 2023Thermodynamic AI and the fluctuation frontiercited by 15
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  7. 2024BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matchingcited by 14
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  9. 2023Entropy-based Training Methods for Scalable Neural Implicit Samplercited by 13
  10. 2021Geometric variational inferencecited by 12
  11. 2021Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Datacited by 12
  12. 2024Amortized Bayesian Multilevel Modelscited by 12
  13. 2022Lagrangian Manifold Monte Carlo on Monge Patchescited by 11
  14. 2022Mutual Information Regularized Offline Reinforcement Learningcited by 11
  15. 2024Variational Inference for Uncertainty Quantification: an Analysis of Trade-offscited by 11
  16. 2024Coding schemes in neural networks learning classification taskscited by 11
  17. 2022Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimizationcited by 10
  18. 2022Alleviating Adversarial Attacks on Variational Autoencoders with MCMCcited by 10
  19. 2022A Flexible Diffusion Modelcited by 10
  20. 2025Microcanonical Langevin Ensembles: Advancing the Sampling of Bayesian Neural Networkscited by 10
  21. 2021Efficient Semi-Implicit Variational Inferencecited by 9
  22. 2021Bayesian graph convolutional neural networks via tempered MCMCcited by 9
  23. 2021Applications of the Free Energy Principle to Machine Learning and Neurosciencecited by 9
  24. 2021Efficient Learning of the Parameters of Non-Linear Models using Differentiable Resampling in Particle Filterscited by 9
  25. 2023Physics-informed Information Field Theory for Modeling Physical Systems with Uncertainty Quantificationcited by 9
  26. 2023GAD-PVI: A General Accelerated Dynamic-Weight Particle-Based Variational Inference Frameworkcited by 9
  27. 2024GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlocited by 9
  28. 2024Dissecting the Interplay of Attention Paths in a Statistical Mechanics Theory of Transformerscited by 9
  29. 2024Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrixcited by 9
  30. 2023Domain constraints improve risk prediction when outcome data is missingcited by 8
  31. 2024Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks: An Extended Investigationcited by 8
  32. 2021Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributionscited by 7
  33. 2022Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Timecited by 7
  34. 2023Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penaltycited by 7
  35. 2023Noise-Free Sampling Algorithms via Regularized Wasserstein Proximalscited by 7
  36. 2024Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadowscited by 7
  37. 2024Randomized Physics-Informed Neural Networks for Bayesian Data Assimilationcited by 7
  38. 2024Importance Corrected Neural JKO Samplingcited by 7

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