On Numerical Considerations for Riemannian Manifold Hamiltonian Monte Carlo

Abstract

Riemannian manifold Hamiltonian Monte Carlo (RMHMC) is a sampling algorithm that seeks to adapt proposals to the local geometry of the posterior distribution. The specific form of the Hamiltonian used in RMHMC necessitates {\it implicitly-defined} numerical integrators in order to sustain reversibility and volume-preservation, two properties that are necessary to establish detailed balance of RMHMC. In practice, these implicit equations are solved to a non-zero convergence tolerance via fixed-point iteration. However, the effect of these convergence thresholds on the ergodicity and computational efficiency properties of RMHMC are not well understood. The purpose of this research is to elucidate these relationships through numerous case studies. Our analysis reveals circumstances wherein the RMHMC algorithm is sensitive, and insensitive, to these convergence tolerances. Our empirical analysis examines several aspects of the computation: (i) we examine the ergodicity of the RMHMC Markov chain by employing statistical methods for comparing probability measures based on collections of samples; (ii) we investigate the degree to which detailed balance is violated by measuring errors in reversibility and volume-preservation; (iii) we assess the efficiency of the RMHMC Markov chain in terms of time-normalized ESS. In each of these cases, we investigate the sensitivity of these metrics to the convergence threshold and further contextualize our results in terms of comparison against Euclidean HMC. We propose a method by which one may select the convergence tolerance within a Bayesian inference application using techniques of stochastic approximation and we examine Newton's method, an alternative to fixed point iterations, which can eliminate much of the sensitivity of RMHMC to the convergence threshold.