# Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods

## TL;DR

This paper introduces Discontinuous Hamiltonian Monte Carlo, an extension of HMC that efficiently samples from distributions with discontinuities, including ordinal parameters, by developing a novel numerical solver that preserves the Hamiltonian exactly.

## Contribution

The paper presents the first numerical solver for discontinuous Hamiltonian dynamics, enabling efficient sampling from complex distributions with discontinuities.

## Key findings

- Effective sampling from discontinuous distributions demonstrated.
- Solver preserves Hamiltonian exactly, ensuring stability.
- Applicable to challenging posterior inference problems.

## Abstract

Hamiltonian Monte Carlo has emerged as a standard tool for posterior computation. In this article, we present an extension that can efficiently explore target distributions with discontinuous densities. Our extension in particular enables efficient sampling from ordinal parameters though embedding of probability mass functions into continuous spaces. We motivate our approach through a theory of discontinuous Hamiltonian dynamics and develop a corresponding numerical solver. The proposed solver is the first of its kind, with a remarkable ability to exactly preserve the Hamiltonian. We apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance.