Positivity of hit-and-run and related algorithms

Abstract

We prove positivity of the Markov operators that correspond to the hit-and-run algorithm, random scan Gibbs sampler, slice sampler and Metropolis algorithm with positive proposal. 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 $MTM^*$, 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 $T$.<br />