# MCMC Using Hamiltonian Dynamics

@article{Neal2011MCMCUH, title={MCMC Using Hamiltonian Dynamics}, author={Radford M. Neal}, journal={arXiv: Computation}, year={2011}, pages={139-188} }

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious "momentum" variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be…

## 2,241 Citations

Stochastic Gradient Hamiltonian Monte Carlo

- Computer ScienceICML
- 2014

A variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution is introduced.

On Irreversible Metropolis Sampling Related to Langevin Dynamics

- Computer ScienceSIAM Journal on Scientific Computing
- 2022

It is shown that as the step size tends to 0, the HAMS proposal satisfies a class of stochastic differential equations including Langevin dynamics as a special case, and theoretical results for HAMS are provided, including algebraic properties of the acceptance probability, the stationary variance, and the expected acceptance rate under a product Gaussian target distribution and the convergence rate under standard Gaussian.

Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo

- Computer Science
- 2022

This work presents the Langevin diﬀusion as an alternative to control these ACFs by inducing randomness in Hamiltonian trajectories through a continuous refreshment of the velocities, and introduces a robust alternative to HMC built upon these dynamics, named Metropolis Adjusted Langevin Trajectories (MALT).

Connecting the Dots: Towards Continuous Time Hamiltonian Monte Carlo

- Mathematics
- 2020

Continuous time Hamiltonian Monte Carlo is introduced, as a powerful alternative to Markov chain Monte Carlo methods for continuous target distributions. The method is constructed in two steps: First…

A splitting Hamiltonian Monte Carlo method for efficient sampling

- Computer ScienceArXiv
- 2021

A splitting Hamiltonian Monte Carlo (SHMC) algorithm, which can be computationally eﬃcient when combined with the random mini-batch strategy, and it is proved that the errors of the Hamiltonian induced by the random batch approximation is O ( √ ∆ t ) in the strong and O (∆ t) in the weak sense.

Hamiltonian Monte Carlo with Energy Conserving Subsampling

- Computer ScienceJ. Mach. Learn. Res.
- 2019

This article shows that efficient subsampling HMC for the parameters is possible if both the dynamics and the acceptance probability are computed from the same data subsample in each complete HMC iteration.

Hamiltonian Dynamics with Non-Newtonian Momentum for Rapid Sampling

- Computer ScienceNeurIPS
- 2021

The proposed Energy Sampling Hamiltonian (ESH) dynamics have a simple form that can be solved with existing ODE solvers, but they derive a specialized solver that exhibits much better performance.

Consistency and Fluctuations For Stochastic Gradient Langevin Dynamics

- Computer ScienceJ. Mach. Learn. Res.
- 2016

This article proves that, under verifiable assumptions, the SGLD algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence (δm)m≥0.

Quasi-Newton Hamiltonian Monte Carlo

- Computer ScienceUAI
- 2016

The theoretical analysis guarantees that this dynamics remains the target distribution invariant and the proposed quasi-Newton Hamiltonian Monte Carlo (QNHMC) algorithm traverses the parameter space more efficiently than the standard HMC and produces a less correlated series of samples.

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