Stein Point Markov Chain Monte Carlo

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.

Minor bug fixed in Theorem 4 (result unchanged)

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• Stochastic Stein Discrepancies
• Optimal quantisation of probability measures using maximum mean discrepancy
• Generalized Kernel Thinning
• Entropy Regularized Motion Planning via Stein Variational Inference
• Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis
• Sampling with Mirrored Stein Operators
• Minimum Discrepancy Methods in Uncertainty Quantification