Hessian-informed Hamiltonian Monte Carlo for high-dimensional problems

TL;DR

This paper explores the use of local Hessian information to enhance Hamiltonian Monte Carlo sampling efficiency in high-dimensional, non-Gaussian Bayesian inference problems, reducing computational complexity.

Contribution

It introduces a Hessian-informed HMC method that uses local Hessian information at each iteration, simplifying the RMHMC approach for high-dimensional problems.

Findings

Local Hessian information improves sampling efficiency.

Hessian-informed HMC outperforms standard methods in high-dimensional tests.

Reduced computational cost compared to full derivative calculations.

Abstract

We investigate the effect of using local and non-local second derivative information on the performance of Hamiltonian Monte Carlo (HMC) sampling methods, for high-dimension non-Gaussian distributions, with application to Bayesian inference and nonlinear inverse problems. The Riemannian Manifold Hamiltonian Monte Carlo (RMHMC) method uses second and third derivative information to improve the performance of the HMC approach. We propose using the local Hessian information at the start of each iteration, instead of re-calculating the higher order derivatives in all sub-steps of the leapfrog updating algorithm. We compare the result of Hessian-informed HMC method using the local and nonlocal Hessian information, in a test bed of a high-dimensional log-normal distribution, related to a problem of inferring soil properties.