Measuring sample quality with diffusions

Abstract

Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target distribution. This approach was recently combined with the theory of reproducing kernels to define a closed-form kernel Stein discrepancy (KSD) computable by summing kernel evaluations across pairs of sample points. We develop a theory of weak convergence for KSDs based on Stein's method, demonstrate that commonly used KSDs fail to detect non-convergence even for Gaussian targets, and show that kernels with slowly decaying tails provably determine convergence for a large class of target distributions. The resulting convergence-determining KSDs are suitable for comparing biased, exact, and deterministic sample sequences and simpler to compute and parallelize than alternative Stein discrepancies. We use our tools to compare biased samplers, select sampler hyperparameters, and improve upon existing KSD approaches to one-sample hypothesis testing and sample quality improvement.

The tamed unadjusted Langevin algorithm

“…[CCG12] and references therein. The existence and regularity of a solution of the Poisson equation has been investigated in [GM96], [PV01], [Kop15], [Gor+16]. In that purpose, the following additional assumption on U is necessary.…”

On Stein’s method for multivariate self-decomposable laws with finite first moment

“…Indeed, the multidimensional Stein's method has mainly been developed for the multivariate normal laws (see e.g. [2,19,20,39,38,11,40,30,33,41,34]) and for invariant measures of multidimensional diffusions ( [28,18]). In particular, the work [18] proposes a general Stein's method framework for target probability measures µ on R d , d ≥ 1, which satisfy the following set of assumptions: µ has finite mean, is absolutely continuous with respect to the d-dimensional Lebesgue measure and its density is continuously differentiable with support the whole of R d .…”

“…Nevertheless, while being very classical in the context of limit theorems, no systematic Stein's method has been implemented for multivariate non-degenerate self-decomposable distributions. (The whole class of non-degenerate self-decomposable laws with finite first moment is different, but intersects with the class of target probability measures considered in [18] and covered by their methodology. Indeed, non-degenerate self-decomposable laws with finite first moment admit a Lebesgue density, which might not be differentiable on R d , and whose support might be a half-space of R d .)…”

On Stein’s method for multivariate self-decomposable laws

“…Originally developed for the Gaussian and the Poisson laws ( [16]), several nonequivalent investigations have focused on extensions and generalizations of Stein's method outside the classical univariate Gaussian and Poisson settings. In this regard, let us cite [7,10,37,43,27,35,38,25,56] and [8,31,29,48,45,15,47,39,40,50,30,2] for univariate and multivariate extensions and generalizations. Moreover, for good introductions to the method, let us refer the reader to the standard references and surveys [24,9,51,18,14].…”