Reversibility of elliptical slice sampling revisited

TL;DR

This paper extends elliptical slice sampling to infinite-dimensional Hilbert spaces, providing new theoretical insights into its reversibility, well-definedness, and the properties of related shrinkage Markov chains.

Contribution

It generalizes elliptical slice sampling to infinite dimensions and offers a new proof of its reversibility and positive semi-definiteness, along with analysis of a related shrinkage chain.

Findings

Extended elliptical slice sampling to infinite-dimensional spaces

Proved reversibility and positive semi-definiteness of the Markov operator

Analyzed a shrinkage Markov chain with potential independent interest

Abstract

We extend elliptical slice sampling, a Markov chain transition kernel suggested in Murray, Adams and MacKay 2010, to infinite-dimensional separable Hilbert spaces and discuss its well-definedness. We point to a regularity requirement, provide an alternative proof of the desirable reversibility property and show that it induces a positive semi-definite Markov operator. Crucial within the proof of the formerly mentioned results is the analysis of a shrinkage Markov chain that may be interesting on its own.