Positivity of hit-and-run and related algorithms

TL;DR

This paper proves the positivity of Markov operators for several algorithms like hit-and-run, Gibbs sampler, slice sampler, and Metropolis, regardless of state space or stationary distribution, simplifying their analysis.

Contribution

It establishes that these algorithms' Markov operators are positive without needing lazy versions, using a novel proof approach based on operator factorization.

Findings

Positivity of hit-and-run, Gibbs, slice sampler, and Metropolis algorithms proven

Positivity is independent of state space and stationary distribution

Lazy versions of these Markov chains are unnecessary

Abstract

We prove positivity of the Markov operators that correspond to the hit-and-run algorithm, random scan Gibbs sampler, slice sampler and an Metropolis algorithm with positive proposal. In all of these cases the positivity is independent of the state space and the stationary distribution. In particular, the results show that it is not necessary to consider the lazy versions of these Markov chains. The proof relies on a well known lemma which relates the positivity of the product M T M^*, for some operators M and T, to the positivity of T. It remains to find that kind of representation of the Markov operator with a positive operator T.