# An Introduction to Hamiltonian Monte Carlo Method for Sampling

@article{Vishnoi2021AnIT, title={An Introduction to Hamiltonian Monte Carlo Method for Sampling}, author={Nisheeth K. Vishnoi}, journal={ArXiv}, year={2021}, volume={abs/2108.12107} }

The goal of this article is to introduce the Hamiltonian Monte Carlo method – a Hamiltonian dynamics inspired algorithm for sampling from a Gibbs density π(x) ∝ e−f(x). We focus on the “idealized” case, where one can compute continuous trajectories exactly. We show that idealized HMC preserves π and we establish its convergence when f is strongly convex and smooth.

## 7 Citations

### On the Dissipation of Ideal Hamiltonian Monte Carlo Sampler

- Physics
- 2022

We report on what seems to be an intriguing connection between variable integration time and partial velocity refreshment of Ideal Hamiltonian Monte Carlo samplers, both of which can be used for…

### Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps

- Computer Science, MathematicsArXiv
- 2022

It is shown that HMC can sample from a distribution that is ε -close in total variation distance using (cid:101) O ( √ κd 1 / 4 log(1 /ε )) gradient queries, where κ is the condition number of Σ.

### Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time

- Computer ScienceArXiv
- 2022

This work proposes a scheme of time-varying integration time based on the roots of Chebyshev polynomials for Hamiltonian Monte Carlo (HMC) and shows that in the case of quadratic potential f, ideal HMC with this choice of integration time only takes O ( √ κ log 1 (cid:15) ) number of iterations to reach Wasserstein-2 distance less than (cids:15).

### A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling

- Computer Science, Mathematics
- 2022

A deterministic particle method for the weighted porous medium equation is developed and its convergence on bounded time intervals is proved and conditions on the target function and data distribution for which convexity of the energy landscape emerges in the continuum limit are identified.

### A blob method method for inhomogeneous diffusion with applications to multi-agent control and sampling

- Computer Science, MathematicsArXiv
- 2022

A deterministic particle method for the weighted porous medium equation is developed and its convergence on bounded time intervals is proved and conditions on the target function and data distribution for which convexity of the energy landscape emerges in the continuum limit are identified.

### Bayesian Inference with Latent Hamiltonian Neural Networks

- Computer ScienceArXiv
- 2022

This work proposes Hamiltonian neural networks (HNNs) with HMC and NUTS for solving Bayesian inference problems, and proposes an HNN extension called latent HNNs (L-Hnns), which is capable of predicting latent variable outputs.

### Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization

- Computer Science, MathematicsArXiv
- 2021

The approach departs from prior works that construct Markov chains on a 1 ε 2 -discretization of K to achieve a sample with ε inﬁnity-distance error, and presents a method to directly convert continuous samples from K with total-variation bounds to samples with in-nity bounds.

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