Measuring Sample Quality with Stein's Method
This work addresses a critical problem for practitioners using biased MCMC procedures, offering a tool for sampler and parameter selection, though it is incremental as it builds on existing Stein's method frameworks.
The paper tackles the challenge of assessing sample quality in biased Markov chain Monte Carlo methods, where traditional measures like effective sample size fail to account for asymptotic bias, by introducing a new computable quality measure based on Stein's method that quantifies discrepancies between sample and target expectations, and demonstrates its utility in applications such as hyperparameter selection and bias-variance tradeoff analysis.
To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. 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 quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.