Fundamentals of Partial Rejection Sampling

TL;DR

This article provides a largely self-contained account of the basic form of the partial rejection sampling algorithm and its analysis, and aims at greater efficiency by resampling only a subset of variables that `go wrong'.

Abstract

Partial Rejection Sampling is an algorithmic approach to obtaining a perfect sample from a specified distribution. The objects to be sampled are assumed to be represented by a number of random variables. In contrast to classical rejection sampling, in which all variables are resampled until a feasible solution is found, partial rejection sampling aims at greater efficiency by resampling only a subset of variables that `go wrong'. Partial rejection sampling is closely related to Moser and Tardos' algorithmic version of the Lovász Local Lemma, but with the additional requirement that a specified output distribution should be met. This article provides a largely self-contained account of the basic form of the algorithm and its analysis.

Figures (5)

• Figure 1: A resampling table
• Figure 2: A realisation of a resampling table, and the corresponding transcript
• Figure 3: The dependency graph $\Gamma$ corresponding to formula $\Phi$ defined in (\ref{['eq:Phi']}).
• Figure 4: A sample graph $\vec{G}$ incorporating a reference orientation
• Figure 5: A pair of clusters in $\mathbb{Z}^2$ with overlapping boundaries

Theorems & Definitions (22)

• Definition 1
• Remark
• Theorem 2
• Lemma 3
• Lemma 4
• proof : Proof of Lemma \ref{['lem:confluence']}
• proof : Proof of Theorem \ref{['thm:PRScorrect']}
• Lemma 5
• Theorem 6
• proof