A Conceptual Introduction to Hamiltonian Monte Carlo

By Michael Betancourt. 2017

1,470 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/59b3ebd49ef620b3051d9777dfc59076d846949f

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Citations in the corpus

Detecting Out-of-Distribution Inputs to Deep Generative Models Using Typicality (2019, methodology): “The figure is inspired by Betancourt [5]’s similar illustration.”
Bayesian Workflow (2020, methodology): “(2019), and Betancourt (2020a). Several of the individual components of Bayesian workflow were discussed by Gelman (2011) but not in a coherent way. In addition there has been development of Bayesian workflow for particular problems, as by Shi and Stevens (2008) and Chiu et al. (2017). In this paper”
Bayesian forecasting in economics and finance: A modern review (2022, background): “Betancourt, M. (2018). A conceptual introduction to Hamiltonian Monte Carlo.”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, methodology): “This generalizes the discussion of Chen et al. (2014) and Betancourt (2015) on Hamiltonian Monte Carlo (HMC) (Duane et al., 1987; Neal et al., 2011; Betancourt, 2017) to general MCMCs.”
ZhuSuan: A Library for Bayesian Deep Learning (2017, background): “The key advantage of HMC is that it exploits information about the geometry of the target distribution (Betancourt, 2017) while only needs the distribution density and the gradient to proceed.”
Likelihood-free MCMC with Approximate Likelihood Ratios (2019, positive): “Hybrid or Hamiltonian Monte Carlo ( HMC ) [14, 4, 15] improves the sampling efﬁciency of vanilla MCMC algorithms by reducing the autocorrelation within the sequence of random Monte Carlo samples.”
A review of automatic differentiation and its efficient implementation (2018, methodology): “Whether or not the above approach is optimal is an open question, most recently posed by (Betancourt, 2017b, 2018), who argues recursively applying AD leads to ineﬃcient and sometimes numerically unstable code.”
Advanced Bayesian Multilevel Modeling with the R Package brms (2017, methodology): “These algorithms converge much more quickly than other Markov-Chain Monte-Carlo (MCMC) algorithms especially for high-dimensional models (Hoffman and Gelman, 2014; Betancourt et al., 2014; Betancourt, 2017).”
dynesty: a dynamic nested sampling package for estimating Bayesian posteriors and evidences (2019, methodology): “In other words, while estimating the posterior mostly depends on generating samples close to where the majority of the distribution is located (i.e. the “typical set”; Betancourt 2017), estimating the evidence requires generating samples in the extended tails of the distribution.”
Rank-Normalization, Folding, and Localization: An Improved Rˆ for Assessing Convergence of MCMC (with Discussion) (2019, methodology): “We observe several divergent transitions and the estimated Bayesian fraction of missing information (Betancourt, 2017) is also low, which indicate convergence problems.”
B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data (2020, methodology): “Hamiltonian Monte Carlo, which is known as a golden approach for sampling from posterior distributions, is an eﬃcient Markov Chain Monte Carlo (MCMC) method based on the Hamiltonian dynamics [15, 16, 23].”
Visualization in Bayesian workflow (2017, background): “But, if we restrict our attention to HMC sampling and its variants, we can obtain much more detailed information about the performance of the Markov chain (Betancourt, 2017).”
Fast Threshold Tests for Detecting Discrimination (2017, background): “The second and third factors depend on the geometry of the posterior: a smooth posterior allows for longer paths between samples that take fewer integration steps to traverse (Betancourt, 2017).”
Bayesian Item Response Modeling in R with brms and Stan (2019, methodology): “The flagship algorithm of Stan is an adaptive Hamiltonian Monte Carlo (HMC) sampler (Betancourt, Byrne, Livingstone, and Girolami 2014; Betancourt 2017; Stan Development Team 2021), which represents a progression from the No-U-Turn Sampler (NUTS) by Hoffman and Gelman (2014).”
Bayesian Variational Autoencoders for Unsupervised Out-of-Distribution Detection (2019, methodology): “We implemented the proposed BVAE model in PyTorch (see Appendix B for pseudocode), using the robust, scale adapted SGHMC variant proposed by Springenberg et al. (2016) 2 .”
Pathfinder: Parallel quasi-Newton variational inference (2021, positive): “…by running many short MCMC chains with dynamic Hamiltonian Monte Carlo in the form of the no-U-turn sampler (Hoffman and Gelman, 2014) as refined by Betancourt (2017).b The short MCMC chains can be viewed as the first stage of warmup for MCMC sampling or as a variational inference algorithm in its…”