The Gibbs sampler revisited from the perspective of conditional modeling

TL;DR

This paper revisits the Gibbs sampler by exploring the equivalence of its foundational operators, introduces the iterative conditional replacement algorithm for convergence analysis, and emphasizes the advantages of operator-specific conditional modeling.

Contribution

It provides a unified perspective on Gibbs sampling operators, proposes the ICR algorithm for convergence proof, and advocates for operator-specific conditional modeling over lumped approaches.

Findings

ICR proves convergence of PCGS with heterogeneous state spaces

The approach simplifies understanding of Gibbs sampler convergence

Operator-specific conditionals outperform lumped conditionals in modeling

Abstract

The Gibbs sampler (GS) is a crucial algorithm for approximating complex calculations, and it is justified by Markov chain theory, the alternating projection theorem, and $I$-projection, separately. We explore the equivalence between these three operators. Partially collapsed Gibbs sampler (PCGS) and pseudo-Gibbs sampler (PGS) are two generalizations of GS. For PCGS, the associated Markov chain is heterogeneous with varying state spaces, and we propose the iterative conditional replacement algorithm (ICR) to prove its convergence. In addition, ICR can approximate the multiple stationary distributions modeled by a PGS. Our approach highlights the benefit of using one operator for one conditional distribution, rather than lumping all the conditionals into one operator. Because no Markov chain theory is required, this approach simplifies the understanding of convergence.