The Polynomial Stein Discrepancy for Assessing Moment Convergence

TL;DR

The paper introduces the polynomial Stein discrepancy (PSD), a scalable and efficient method for assessing moment convergence and sample quality in Bayesian inference, outperforming existing techniques in power and computational cost.

Contribution

It develops the PSD and a related goodness-of-fit test that detects moment differences, addressing limitations of kernel Stein discrepancy methods.

Findings

The PSD-based test detects differences in the first r moments for Gaussian targets.

The PSD test has higher power than competitors in several examples.

The PSD aids in hyper-parameter selection for Bayesian sampling algorithms.

Abstract

We propose a novel method for measuring the discrepancy between a set of samples and a desired posterior distribution for Bayesian inference. Classical methods for assessing sample quality like the effective sample size are not appropriate for scalable Bayesian sampling algorithms, such as stochastic gradient Langevin dynamics, that are asymptotically biased. Instead, the gold standard is to use the kernel Stein Discrepancy (KSD), which is itself not scalable given its quadratic cost in the number of samples. The KSD and its faster extensions also typically suffer from the curse of dimensionality and can require extensive tuning. To address these limitations, we develop the polynomial Stein discrepancy (PSD) and an associated goodness-of-fit test. While the new test is not fully convergence-determining, we prove that it detects differences in the first r moments for Gaussian targets. We empirically show that the test has higher power than its competitors in several examples, and at a lower computational cost. Finally, we demonstrate that the PSD can assist practitioners to select hyper-parameters of Bayesian sampling algorithms more efficiently than competitors.