Stein Point Markov Chain Monte Carlo

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

A Riemannian-Stein Kernel Method

This paper presents a theoretical analysis of numerical integration based on interpolation with a Stein kernel. In particular, the case of integrals with respect to a posterior distribution supported on a general Riemannian manifold is considered and the asymptotic convergence of the estimator in this context is established. Our results are considerably stronger than those previously reported, in that the optimal rate of convergence is established under a basic Sobolev-type assumption on the integrand. The theoretical results are empirically verified on $\mathbb{S}^2$

Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, remain the gold standard for approximate Bayesian inference because they have a robust finite-sample theory and reliable convergence diagnostics. However, alternative methods, which are more scalable or apply to problems where Markov Chain Monte Carlo cannot be used, lack the same finite-data approximation theory and tools for evaluating their accuracy. In this work, we develop a flexible new approach to bounding the error of mean and uncertainty estimates of scalable inference algorithms. Our strategy is to control the estimation errors in terms of Wasserstein distance, then bound the Wasserstein distance via a generalized notion of Fisher distance. Unlike computing the Wasserstein distance, which requires access to the normalized posterior distribution, the Fisher distance is tractable to compute because it requires access only to the gradient of the log posterior density. We demonstrate the usefulness of our Fisher distance approach by deriving bounds on the Wasserstein error of the Laplace approximation and Hilbert coresets. We anticipate that our approach will be applicable to many other approximate inference methods such as the integrated Laplace approximation, variational inference, and approximate Bayesian computationComment: 22 pages, 2 figure