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Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian…

Machine Learning · Statistics 2022-09-27 Simon Apers , Sander Gribling , Dániel Szilágyi

Hamiltonian Monte Carlo (HMC) is a state of the art method for sampling from distributions with differentiable densities, but can converge slowly when applied to challenging multimodal problems. Running HMC with a time varying Hamiltonian,…

Machine Learning · Statistics 2026-02-26 Reuben Cohn-Gordon , Uroš Seljak , Dries Sels

We here investigate the efficient implementation of the energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced for the numerical solution of Hamiltonian problems. In this note, we describe an…

Numerical Analysis · Mathematics 2013-10-22 Luigi Brugnano , Gianluca Frasca Caccia , Felice Iavernaro

Recently, the Hamilton Monte Carlo (HMC) has become widespread as one of the more reliable approaches to efficient sample generation processes. However, HMC is difficult to sample in a multimodal posterior distribution because the HMC chain…

Computation · Statistics 2020-06-22 Jonghyun Yun , Minsuk Shin , Ick Hoon Jin , Faming Liang

We introduce a Hamiltonian Monte Carlo (HMC) methodology based on a randomized selection of integration times, referred to as eHMC, where "e" stands for empirical. The approach relies on an offline calibration phase that leverages…

Computation · Statistics 2026-05-25 Changye Wu , Pierre Pudlo , Christian P. Robert , Julien Stoehr

Hamiltonian Monte Carlo (HMC) is widely used for sampling from high dimensional target distributions with densities known up to proportionality. While HMC exhibits favorable scaling properties in high dimensions, it struggles with strongly…

Computation · Statistics 2025-07-30 Joonha Park

A novel computationally efficient Markov chain Monte Carlo (MCMC) scheme for latent Gaussian models (LGMs) is proposed in this paper. The sampling scheme is a two block Gibbs sampling scheme designed to exploit the model structure of LGMs.…

Computation · Statistics 2015-06-23 Óli Páll Geirsson , Birgir Hrafnkelsson , Daniel Simpson , Helgi Sigurðarson

We introduce the energy-stepping Monte Carlo (ESMC) method, a Markov chain Monte Carlo (MCMC) algorithm based on the conventional dynamical interpretation of the proposal stage but employing an energy-stepping integrator. The…

Mathematical Physics · Physics 2023-12-13 Ignacio Romero , Michael Ortiz

Hamiltonian Monte Carlo (HMC) is a popular method in sampling. While there are quite a few works of studying this method on various aspects, an interesting question is how to choose its integration time to achieve acceleration. In this…

Machine Learning · Computer Science 2023-02-16 Jun-Kun Wang , Andre Wibisono

We present a new Subset Simulation approach using Hamiltonian neural network-based Monte Carlo sampling for reliability analysis. The proposed strategy combines the superior sampling of the Hamiltonian Monte Carlo method with…

Machine Learning · Statistics 2024-01-11 Denny Thaler , Somayajulu L. N. Dhulipala , Franz Bamer , Bernd Markert , Michael D. Shields

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as…

Probability · Mathematics 2020-07-21 Nawaf Bou-Rabee , Jesús María Sanz-Serna

The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…

Statistics Theory · Mathematics 2017-12-15 Radislav Vaisman , Robert Salomone , Dirk P. Kroese

This paper studies a non-random-walk Markov Chain Monte Carlo method, namely the Hamiltonian Monte Carlo (HMC) method in the context of Subset Simulation used for structural reliability analysis. The HMC method relies on a deterministic…

Computation · Statistics 2018-04-20 Ziqi Wang , Marco Broccardo , Junho Song

We discuss systematic extensions of the standard (St{\"o}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics, with relative accuracy of order $\tau^2$ for a timestep of length $\tau$, to higher orders in…

Numerical Analysis · Mathematics 2013-10-09 Asif Mushtaq , Anne Kværnø , Kåre Olaussen

We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard…

Numerical Analysis · Mathematics 2021-06-25 S. Blanes , M. P. Calvo , F. Casas , J. M. Sanz-Serna

Hamiltonian splitting methods are an established technique to derive stable and accurate integration schemes in molecular dynamics, in which additional accuracy can be gained using force gradients. For rigid bodies, a tradition exists in…

Statistical Mechanics · Physics 2008-04-10 Ramses van Zon , Igor P. Omelyan , Jeremy Schofield

We introduce Markov chain Monte Carlo (MCMC) algorithms based on numerical approximations of piecewise-deterministic Markov processes obtained with the framework of splitting schemes. We present unadjusted as well as adjusted algorithms,…

Probability · Mathematics 2025-11-04 Andrea Bertazzi , Paul Dobson , Pierre Monmarché

Hamiltonian Monte Carlo (HMC) is a popular Markov Chain Monte Carlo (MCMC) algorithm to sample from an unnormalized probability distribution. A leapfrog integrator is commonly used to implement HMC in practice, but its performance can be…

Computation · Statistics 2021-10-28 Marcel Hirt , Michalis K. Titsias , Petros Dellaportas

Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) method for performing approximate inference in complex probabilistic models of continuous variables. In common with many MCMC methods, however, the standard HMC…

Computation · Statistics 2017-04-12 Matthew M. Graham , Amos J. Storkey

Gradient-based Monte Carlo sampling algorithms, like Langevin dynamics and Hamiltonian Monte Carlo, are important methods for Bayesian inference. In large-scale settings, full-gradients are not affordable and thus stochastic gradients…

Machine Learning · Computer Science 2019-06-25 Zhize Li , Tianyi Zhang , Shuyu Cheng , Jun Zhu , Jian Li