Measuring Sample Quality with Stein’s Method

To carry out posterior inference on datasets of unprecedented sizes, practitioners are turning to biased MCMC procedures that trade off asymptotic exactness for computational efficiency. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein’s method that bounds the discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, asymptotically biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.