Measuring Sample Quality with Diffusions

Standard Markov chain Monte Carlo diagnostics, like effective sample size, are ineffective for biased sampling procedures that sacrifice asymptotic correctness for computational speed. Recent work addresses this issue for a class of strongly log-concave target distributions by constructing a computable discrepancy measure based on Stein's method that provably determines convergence to the target. We generalize this approach to cover any target with a fast-coupling Ito diffusion by bounding the derivatives of Stein equation solutions in terms of Markov process coupling times. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyperparameters of biased samplers, compare random and deterministic quadrature rules, and quantify bias-variance tradeoffs in approximate Markov chain Monte Carlo. Our explicit multivariate Stein factor bounds may be of independent interest.