Measuring Sample Quality with Diffusions
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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.
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