Measuring Sample Quality with Diffusions.

Stein's method for measuring convergence to a continuous target distributionrelies on an operator characterizing the target and Stein factor bounds on thesolutions of an associated differential equation. While such operators andbounds are readily available for a diversity of univariate targets, fewmultivariate targets have been analyzed. We introduce a new class ofcharacterizing operators based on Ito diffusions and develop explicitmultivariate Stein factor bounds for any target with a fast-coupling Itodiffusion. As example applications, we develop computable andconvergence-determining diffusion Stein discrepancies for log-concave,heavy-tailed, and multimodal targets and use these quality measures to selectthe hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, comparerandom and deterministic quadrature rules, and quantify bias-variance tradeoffsin approximate MCMC. Our results establish a near-linear relationship betweendiffusion Stein discrepancies and Wasserstein distances, improving upon pastwork even for strongly log-concave targets. The exposed relationship betweenStein factors and Markov process coupling may be of independent interest.