Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials

TL;DR

This paper provides precise convergence rate estimates for Gibbs sampling in simple models, especially focusing on cases where standard techniques are ineffective, and extends to bivariate and Lancaster family models.

Contribution

It introduces sharper convergence bounds for Gibbs sampler in specific models, advancing understanding of its efficiency in complex distributions.

Findings

Derived sharp convergence rates for Gibbs sampler

Extended analysis to bivariate models and Lancaster families

Identified limitations of existing bounding techniques

Abstract

We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there. [arXiv:0808.3852]