Measuring Sample Quality with Kernels

TL;DR

This paper develops a kernel Stein discrepancy (KSD) framework to reliably detect convergence of samples to target distributions, addressing limitations of existing diagnostics and enabling improved sample quality assessment.

Contribution

It introduces a new theory of weak convergence for KSDs, identifies limitations of current KSDs, and proposes kernels with slowly decaying tails for better convergence detection.

Findings

Common KSDs fail to detect non-convergence for Gaussian targets

Kernels with slowly decaying tails ensure convergence detection for many distributions

KSDs are effective for comparing biased and deterministic samples

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.