Stein Point Markov Chain Monte Carlo
This addresses the computational bottleneck in Bayesian inference and machine learning for researchers and practitioners, offering a more efficient method for probability approximation.
The paper tackled the computational inefficiency of Stein Points algorithms for approximating probability measures by replacing the non-convex optimization step with Markov chain sampling, reducing computational cost and enabling straightforward implementation, as demonstrated on challenging Bayesian inference problems with consistency guarantees.
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.