Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reversibility of elliptical slice sampling revisited

Mareike Hasenpflug, Viacheslav Telezhnikov, Daniel Rudolf

TL;DR

The paper addresses whether elliptical slice sampling (ESS) remains well-defined and stationary when the target distribution lives on an infinite-dimensional Hilbert space, with the posterior $\mu(d x)=Z^{-1}\rho(x)\mu_0(d x)$ and $\mu_0=\mathcal{N}(0,C)$. It develops a circle-angle shrinkage representation, proves termination under the mild condition that $\rho$ is lower semicontinuous, and establishes that the ESS transition kernel is reversible w.r.t. $\mu$ and that the induced operator on $L^2_\mu$ is positive semi-definite. The key technical tool is the stopped shrinkage kernel $Q_S$ on open-on-circle sets $S$, together with a representation of the ESS update as a mixture over angles; these yield a pathway to dimension-independent mixing and motivate spectral-gap analyses and extensions to approximate ESS and related slice-sampling variants. The results advance the theoretical understanding of slice sampling on function spaces and suggest practical directions for coupling constructions and broadened applicability.

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.

Reversibility of elliptical slice sampling revisited

TL;DR

The paper addresses whether elliptical slice sampling (ESS) remains well-defined and stationary when the target distribution lives on an infinite-dimensional Hilbert space, with the posterior and . It develops a circle-angle shrinkage representation, proves termination under the mild condition that is lower semicontinuous, and establishes that the ESS transition kernel is reversible w.r.t. and that the induced operator on is positive semi-definite. The key technical tool is the stopped shrinkage kernel on open-on-circle sets , together with a representation of the ESS update as a mixture over angles; these yield a pathway to dimension-independent mixing and motivate spectral-gap analyses and extensions to approximate ESS and related slice-sampling variants. The results advance the theoretical understanding of slice sampling on function spaces and suggest practical directions for coupling constructions and broadened applicability.

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.
Paper Structure (10 sections, 15 theorems, 104 equations, 1 figure, 3 algorithms)

This paper contains 10 sections, 15 theorems, 104 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Properties and notation of the shrinkage procedure
  4. Stopping of the shrinkage procedure
  5. Properties of the stopped shrinkage procedure
  6. Reversibility and positive semi-definiteness of elliptical slice sampling
  7. Summary and outlook
  8. Technical proofs
  9. Proof of Lemma \ref{['lem: theta_conditioned_on_interval']}
  10. Proof of Lemma \ref{['lem: pushforward_invariance']}

Key Result

Lemma 2.1

Let $\alpha, \beta, \gamma \in [0,2\pi)$. If $\gamma \neq \alpha$ we have $\mathbf{1}_{I(\alpha,\beta)}(\gamma) = \mathbf{1}_{J(\beta,\gamma)}(\alpha).$ Conversely if $\gamma \neq \beta$ we have $\mathbf{1}_{J(\alpha,\beta)}(\gamma) = \mathbf{1}_{I(\gamma,\alpha)}(\beta).$

Figures (1)

  • Figure 1: The dependency graph of the $(Z_n)_{n\in\mathbb{N}}$ conditioned on $\Theta=\theta$.

Theorems & Definitions (31)

  • Lemma 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 21 more