# Kinetic energy choice in Hamiltonian/hybrid Monte Carlo

## TL;DR

This paper investigates how different kinetic energy choices in Hamiltonian Monte Carlo influence performance, revealing that standard Gaussian momentum may not always be optimal and highlighting a trade-off between efficiency and robustness.

## Contribution

It introduces two evaluative quantities for kinetic energy choices and analyzes their impact on integrator stability, convergence, and the undesirable negligible moves property.

## Key findings

- Heavy-tailed momentum distributions can cause negligible moves.
- Standard Gaussian momentum is not always optimal.
- A trade-off exists between efficiency and robustness.

## Abstract

We consider how different choices of kinetic energy in Hamiltonian Monte Carlo affect algorithm performance. To this end, we introduce two quantities which can be easily evaluated, the composite gradient and the implicit noise. Results are established on integrator stability and geometric convergence, and we show that choices of kinetic energy that result in heavy-tailed momentum distributions can exhibit an undesirable negligible moves property, which we define. A general efficiency-robustness trade off is outlined, and implementations which rely on approximate gradients are also discussed. Two numerical studies illustrate our theoretical findings, showing that the standard choice which results in a Gaussian momentum distribution is not always optimal in terms of either robustness or efficiency.