Stein Point Markov Chain Monte Carlo

TL;DR

This paper introduces a new class of algorithms called Stein Point Markov Chain Monte Carlo that efficiently approximate probability measures by leveraging Markov chain sampling, reducing computational costs and maintaining theoretical guarantees.

Contribution

It replaces non-convex optimization in Stein Points with Markov chain sampling, simplifying implementation and improving efficiency.

Findings

Reduced computational cost compared to traditional Stein Points

Algorithms successfully applied to complex Bayesian inference problems

Established theoretical guarantees of consistency

Abstract

An important task in machine learning and statistics is the approximation of a probability measure by an empirical measure supported on a discrete point set. Stein Points are a class of algorithms for this task, which proceed by sequentially minimising a Stein discrepancy between the empirical measure and the target and, hence, require the solution of a non-convex optimisation problem to obtain each new point. This paper removes the need to solve this optimisation problem by, instead, selecting each new point based on a Markov chain sample path. This significantly reduces the computational cost of Stein Points and leads to a suite of algorithms that are straightforward to implement. The new algorithms are illustrated on a set of challenging Bayesian inference problems, and rigorous theoretical guarantees of consistency are established.