Rates of convergence for Gibbs sampling in the analysis of almost exchangeable data

TL;DR

This paper analyzes the convergence rate of a Gibbs sampling method for a class of probability distributions related to almost exchangeable data, showing it mixes in quadratic time relative to a parameter A.

Contribution

It provides explicit bounds on the mixing time of the Gibbs sampler for these distributions, depending on spectral properties of the weight matrix.

Findings

Mixing time is Θ(A^2) steps for large A.

Bounds depend on spectral parameters of the matrix C.

Results apply to distributions motivated by de Finetti's theorem.

Abstract

Motivated by de Finetti's representation theorem for almost exchangeable arrays, we want to sample $\mathbf p \in [0,1]^d$ from a distribution with density proportional to $\exp(-A^2\sum_{i<j}c_{ij}(p_i-p_j)^2)$, where $A$ is large and $c_{ij}$'s are non-negative weights. We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every non-zero fixed matrix $C=(c_{ij})$, and large enough $A$, mixing happens in $\Theta(A^2)$ steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix $C$ through few relevant spectral parameters.