Measuring Sample Quality with Kernels
This work addresses the challenge of ensuring reliable inference in approximate MCMC for researchers in statistics and machine learning, offering a practical and theoretically grounded method for sample quality assessment.
The paper tackled the problem of detecting biases in approximate Markov chain Monte Carlo (MCMC) sampling by developing kernel Stein discrepancies (KSDs) that provably determine convergence to target distributions, showing that certain kernels with slowly decaying tails are effective and applying these tools to compare samplers and improve sample quality.
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.