On the convergence rate of the "out-of-order" block Gibbs sampler

TL;DR

This paper investigates how reordering steps in a block Gibbs sampler affects its convergence and invariant distribution, revealing that while the reordering can invalidate the algorithm, the convergence rate remains unchanged.

Contribution

It demonstrates that out-of-order reordering can invalidate the invariant distribution but preserves the convergence rate, providing insights into the analysis of Gibbs samplers.

Findings

Out-of-order Gibbs sampler does not have the correct invariant distribution.

Convergence rate remains the same despite reordering.

Both chains are either geometrically ergodic or not, simultaneously.

Abstract

It is shown that a seemingly harmless reordering of the steps in a block Gibbs sampler can actually invalidate the algorithm. In particular, the Markov chain that is simulated by the "out-of-order" block Gibbs sampler does not have the correct invariant probability distribution. However, despite having the wrong invariant distribution, the Markov chain converges at the same rate as the original block Gibbs Markov chain. More specifically, it is shown that either both Markov chains are geometrically ergodic (with the same geometric rate of convergence), or neither one is. These results are important from a practical standpoint because the (invalid) out-of-order algorithm may be easier to analyze than the (valid) block Gibbs sampler (see, e.g., Yang and Rosenthal [2019]).