Gibbs Sampling, Exponential Families and Orthogonal Polynomials

TL;DR

This paper provides explicit examples of Gibbs samplers with sharp convergence rates, using exponential families and orthogonal polynomials to analyze their spectral properties and convergence behavior.

Contribution

It introduces a class of Gibbs samplers with explicitly diagonalizable transition operators using classical orthogonal polynomials, enabling precise convergence analysis.

Findings

Explicit convergence rates for Gibbs samplers in exponential families

Diagonalization of transition operators using orthogonal polynomials

Identification of eigenfunctions for spectral analysis

Abstract

We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.