Hamiltonian Monte Carlo in Inverse Problems; Ill-Conditioning and Multi-Modality

TL;DR

This paper analyzes the challenges of using Hamiltonian Monte Carlo in inverse problems, focusing on poor conditioning and multi-modality, and offers practical recommendations for improved sampling in spectroscopy and plasma reconstruction.

Contribution

It introduces novel preconditioning and replica exchange strategies for HMC tailored to inverse problems, with rigorous analysis and practical guidelines.

Findings

Preconditioning improves HMC sampling efficiency.

Replica exchange enhances exploration of multi-modal distributions.

Guidelines for step size, integration steps, and annealing improve sampling performance.

Abstract

The Hamiltonian Monte Carlo (HMC) method allows sampling from continuous densities. Favorable scaling with dimension has led to wide adoption of HMC by the statistics community. Modern auto-differentiating software should allow more widespread usage in Bayesian inverse problems. This paper analyzes two major difficulties encountered using HMC for inverse problems: poor conditioning and multi-modality. Novel results on preconditioning and replica exchange Monte Carlo parameter selection are presented in the context of spectroscopy. Recommendations are given for the number of integration steps as well as step size, preconditioner type and fitting, annealing form and schedule. These recommendations are analyzed rigorously in the Gaussian case, and shown to generalize in a fusion plasma reconstruction.