On Hamiltonian Monte Carlo for Gaussian Random Variables with Random Hamiltonians

TL;DR

This paper analyzes Gaussian Hamiltonian Monte Carlo operators, proving invariance of Gaussian distributions, contraction properties, and deriving explicit formulas to understand their dynamics and convergence behavior.

Contribution

It introduces a family of GHMC operators, proves their invariance for Gaussian distributions, and provides explicit formulas for their contraction properties and dynamics.

Findings

Gaussian distributions are invariant under GHMC operators

GHMC operators are contractions on parameter space

Explicit formulas for GHMC operator dynamics are derived

Abstract

We study a family of (multivariate-)Gaussian Hamiltonian Monte Carlo (GHMC) operators and prove that the family of Gaussian distributions and their mixtures are invariant under such operators. Furthermore, each such operator is a contraction on the space of parameters and an explicit formulae are derived. These results then enable us to analyze the dynamics and convergences of independent and identically distributed random sequences of such operators.