The Geometric Foundations of Hamiltonian Monte Carlo

By Michael Betancourt, Simon Byrne, Samuel Livingstone, Mark A. Girolami. 2014

197 citations recorded by the source corpus.

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory naturally identifies efficient implementations and motivates promising generalizations.

Subfields: Mathematical Physics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/05c67f2e471d77953d016318bbf49c7c145a6df3

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Citations in the corpus

A Conceptual Introduction to Hamiltonian Monte Carlo (2017, methodology): “Instead of relying on fragile heuristics, the method is built upon a rich theoretical foundation that makes it uniquely suited to the highdimensional problems of applied interest (Betancourt et al., 2014).”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “For a complete account of this important ﬂavour of MCMC, the reader is referred to Neal (2013), which inspired the description below; see also Betancourt et al. 2014 for a highly mathematical diﬀerential-geometry approach to HMC.”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “This could encourage the dynamics to explore a broader area to reduce sample autocorrelation and improve efficiency (e.g. Neal et al. (2011); Ding et al. (2014); Betancourt et al. (2017)).”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, methodology): “Hamiltonian MCMC (see review of HMC in [1, 2]) has been empirically observed to converge faster to the invariant distribution compared to standard Langevin MCMC which is a discretization of overdamped Langevin diffusion (the first order SDE corresponding to the high friction limit of (1)).”
Generalized Energy Based Models (2020, methodology): “Kinetic samplers were shown to better explore the modes of the invariant distribution ν compared to Overdamped ones (see (Neal, 2010; Betancourt et al., 2017) for empirical results and (Cheng et al., 2017) for theory), as also confirmed empirically in Appendix D for image generation tasks using…”
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).”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, background): “What drives the exploration of the different values of H (θ, p ) is therefore the simulation of the momentum, which makes its calibration both quite inﬂuential and delicate (Betancourt et al. 2014) as it depends on the unknown normalising constant of the target.”
The Stan Math Library: Reverse-Mode Automatic Differentiation in C++ (2015, methodology): “For MCMC, Hamiltonian Monte Carlo (HMC) (Duane et al., 1987; Neal, 2011; Betancourt et al., 2014) requires gradients in the leapfrog integrator it uses to solve the Hamiltonian equations and thus simulate an efficient flow through the posterior.”
On the convergence of Hamiltonian Monte Carlo (2017, methodology): “We refer the interested readers to the monograph [13] and the surveys given in [14, Chapter 9], [21], [3] and [4].”
A Markov Jump Process for More Efficient Hamiltonian Monte Carlo (2015, methodology): “These include the use of shadow Hamiltonians that are more closely conserved by the approximate Hamiltonian integrator [11], Riemann manifold HMC [12] and other investigation of its geometry [13], quasi-Newton HMC [14], Hilbert space HMC [15], parameter adapta-”
Optimizing The Integrator Step Size for Hamiltonian Monte Carlo (2014, background): “Hamiltonian Monte Carlo (Duane et al., 1987; Neal, 2011; Betancourt et al., 2014) yields efficient inference that scales to high-dimensional problems by building up Markov transitions from a smooth maps known as Hamiltonian flow.”