A Gibbs sampler on the $n$-simplex

TL;DR

This paper analyzes the mixing time of a Gibbs sampler on the n-simplex, confirming Aldous's conjecture, and introduces a perfect sampling algorithm based on their proof for more complex Gibbs samplers.

Contribution

It provides a rigorous determination of the mixing time for a Gibbs sampler on the n-simplex and presents a novel perfect sampling algorithm applicable to complex cases.

Findings

Confirmed Aldous's conjecture on mixing time.

Developed a two-step coupling method for analysis.

Proposed a MCMC-based perfect sampling algorithm.

Abstract

We determine the mixing time of a simple Gibbs sampler on the unit simplex, confirming a conjecture of Aldous. The upper bound is based on a two-step coupling, where the first step is a simple contraction argument and the second step is a non-Markovian coupling. We also present a MCMC-based perfect sampling algorithm based on our proof which can be applied with Gibbs samplers that are harder to analyze.