On Gibbs Sampling for Structured Bayesian Models Discussion of paper by Zanella and Roberts

Abstract

We congratulate Professors Giacomo Zanella and Gareth Roberts for their path-breaking work in analyzing Gibbs sampling algorithms for a class of highly practical Bayesian hierarchical models. Together with their previous work, Papaspiliopoulos and Roberts (2003) and Papaspiliopoulos et al. (2020), their multigrid decomposition strategy elegantly reduces a high-dimensional Gibbs sampling algorithm to independent low-dimensional components so that the convergence rate of the Gibbs sampler can be determined analytically.These are extremely interesting and encouraging results. Throughout of the article, we will refer to this work of Zanella and Roberts (2021) as "Z&R" for simplicity.The multigrid decomposition serves a central role in the whole theory established in the aforementioned series of papers. An intuition behind this decomposition is that lower-level mean statistics are sufficient for posterior inference on upper-level parameters, with lowerlevel parameters practically marginalized out. For example, Papaspiliopoulos and Roberts * This paper was done while Xiaodong Yang was working as a remote summer undergraduate at Professor Jun S. Liu's lab at Harvard, in the summer of 2021.