Fast Approximate Solution of Stein Equations for Post-Processing of MCMC

TL;DR

This paper introduces a fast, approximate method for solving Stein equations in Bayesian inference, reducing computational costs by combining iterative solvers and preconditioning, thus enabling efficient posterior expectation calculations.

Contribution

It proposes a novel approach that accelerates Stein equation solutions using iterative methods and preconditioning, improving efficiency over existing high-cost numerical techniques.

Findings

Significant reduction in computational time for Stein equation solutions.

Effective use of iterative solvers with preconditioning strategies.

Potential for scalable Bayesian inference with smoother integrands.

Abstract

Bayesian inference is conceptually elegant, but calculating posterior expectations can entail a heavy computational cost. Monte Carlo methods are reliable and supported by strong asymptotic guarantees, but do not leverage smoothness of the integrand. Solving Stein equations has emerged as a possible alternative, providing a framework for numerical approximation of posterior expectations in which smoothness can be exploited. However, existing numerical methods for Stein equations are associated with high computational cost due to the need to solve large linear systems. This paper considers the combination of iterative linear solvers and preconditioning strategies to obtain fast approximate solutions of Stein equations.