# Hug and Hop: a discrete-time, non-reversible Markov chain Monte-Carlo algorithm

## TL;DR

The paper introduces the Hug and Hop MCMC algorithm, a novel non-reversible Markov chain method that efficiently estimates expectations for complex distributions, often outperforming Hamiltonian Monte Carlo.

## Contribution

It presents a new non-reversible MCMC algorithm combining bounce-based proposals with contour-jumping, leveraging local Hessian info without implicit steps.

## Key findings

- Hug and Hop often outperform Hamiltonian Monte Carlo in experiments.
- The algorithm maintains high acceptance rates with high-dimensional targets.
- It is robust to unbounded gradients of the log-posterior.

## Abstract

We introduced the Hug and Hop Markov chain Monte Carlo algorithm for estimating expectations with respect to an intractable distribution. The algorithm alternates between two kernels: Hug and Hop. Hug is a non-reversible kernel that repeatedly applies the bounce mechanism from the recently proposed Bouncy Particle Sampler to produce a proposal point far from the current position, yet on almost the same contour of the target density, leading to a high acceptance probability. Hug is complemented by Hop, which deliberately proposes jumps between contours and has an efficiency that degrades very slowly with increasing dimension. There are many parallels between Hug and Hamiltonian Monte Carlo using a leapfrog integrator, including the order of the integration scheme, however Hug is also able to make use of local Hessian information without requiring implicit numerical integration steps, and its performance is not terminally affected by unbounded gradients of the log-posterior. We test Hug and Hop empirically on a variety of toy targets and real statistical models and find that it can, and often does, outperform Hamiltonian Monte Carlo.