Title: Measuring Sample Quality with Kernels,
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.