Convergence rates of the random scan Gibbs sampler under the Dobrushin’s uniqueness condition

Neng-Yi Wang

doi:10.1214/17-ecp90

Abstract

In this paper, under the Dobrushin’s uniqueness condition, we obtain explicit estimates of the geometrical convergence rate for the random scan Gibbs sampler in the Wasserstein metric.

Highlights

Let μ be a Gibbs probability measure on EN with dimension N, i.e., μ(dx1, · · ·, dxN ) =EN e−V e−V (x1,···,xN ) (x1,···,xN )π(dx1) · ·· π(dx1) π(dxN )· · π(dxN ), where π is some σ-finite reference measure on E
E−V (x1,···,xN ) e−V (x1,···,xN )π(dxi) π(dxi), which is a one-dimensional measure, easy to simulate in practice
The scheme of the random scan Gibbs sampler approximating μ is that, in each iteration, one randomly chooses one coordinate to update according to the one-dimensional conditional distributions μi, i = 1, · · ·, N

Summary

Introduction

In order to approximate μ via iterations of the one-dimensional conditional distributions μi, i = 1, · · · , N, the various scan Gibbs samplers are often used (see [4]). In [6], Wu and the author studied systematic scan Gibbs sampler by Dobrushin’s uniqueness conditions. The scheme of the random scan Gibbs sampler approximating μ is that, in each iteration, one randomly chooses one coordinate to update according to the one-dimensional conditional distributions μi, i = 1, · · · , N.

Main results

Proofs of the main results