# A Conceptual Introduction to Hamiltonian Monte Carlo

@article{Betancourt2017ACI, title={A Conceptual Introduction to Hamiltonian Monte Carlo}, author={Michael Betancourt}, journal={arXiv: Methodology}, year={2017} }

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…

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## 691 Citations

Geometry and Dynamics for Markov Chain Monte Carlo

- MathematicsArXiv
- 2017

The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners, and other users of the methodology with only a basic understanding of Monte Carlo methods.

Geometry & Dynamics for Markov Chain Monte Carlo

- Computer Science
- 2021

The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners and other users of the methodology with only a basic understanding of Monte Carlo methods.

Unbiased Hamiltonian Monte Carlo with couplings

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- 2019

The choice of algorithmic parameters and the efficiency of the proposed approach are illustrated on a logistic regression with 300 covariates and a log-Gaussian Cox point processes model with low- to fine-grained discretizations.

Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods

- Computer Science, Mathematics
- 2020

An extension of Hamiltonian Monte Carlo that can efficiently explore target distributions with discontinuous densities and enables efficient sampling from ordinal parameters though embedding of probability mass functions into continuous spaces is presented.

Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions

- Computer Science
- 2019

This work provides intuition as well as a formal analysis showing how these multiscale distributions limit the stepsize of leapfrog and shows how the implicit midpoint method can be used, together with Newton-Krylov iteration, to circumvent this limitation and achieve major efficiency gains.

Probabilistic Path Hamiltonian Monte Carlo

- MathematicsICML
- 2017

Probabilistic Path HMC (PPHMC) is developed as a first step to sampling distributions on spaces with intricate combinatorial structure, and a surrogate function to ease the transition across a boundary on which the log-posterior has discontinuous derivatives can greatly improve efficiency.

Magnetic Hamiltonian Monte Carlo

- Physics, MathematicsICML
- 2017

A theoretical basis for the use of non-canonical Hamiltonian dynamics in MCMC is established, and a symplectic, leapfrog-like integrator is constructed allowing for the implementation of magnetic HMC.

A Condition Number for Hamiltonian Monte Carlo.

- Computer Science
- 2019

This paper restricts attention to multivariate Gaussian targets, and obtains a condition number corresponding to sampling efficiency, which quantifies the number of leapfrog steps needed to efficiently sample.

Several Remarks on the Numerical Integrator in Lagrangian Monte Carlo

- Computer Science
- 2022

This work makes several contributions regarding the numerical integrator used in LMC, and demonstrates that the LMC integrator enjoys a certain robustness to human error that is not shared with the generalized leapfrog integrator, which can invalidate detailed balance in the latter case.

Optimization algorithms inspired by the geometry of dissipative systems

- Computer Science
- 2019

Dynamical systems defined through a contact geometry are introduced which are not only naturally suited to the optimization goal but also subsume all previous methods based on geometric dynamical systems, which shows that optimization algorithms that achieve oracle lower bounds on convergence rates can be obtained.

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