Generating MCMC proposals by randomly rotating the regular simplex

TL;DR

The paper introduces the simplicial sampler, a parallel MCMC method that uses random rotations of a simplex to generate proposals, simplifying acceptance steps and improving efficiency.

Contribution

It proposes a novel simplex-based multiproposal mechanism for MCMC that simplifies acceptance and enhances parallel efficiency, with theoretical and practical speedups.

Findings

Simplified acceptance step based on simplex rotations

Effective parallelization across dimensions and distributions

Speed improvements demonstrated in experiments

Abstract

We present the simplicial sampler, a class of parallel MCMC methods that generate and choose from multiple proposals at each iteration. The algorithm's multiproposal randomly rotates a simplex connected to the current Markov chain state in a way that inherently preserves symmetry between proposals. As a result, the simplicial sampler leads to a simplified acceptance step: it simply chooses from among the simplex nodes with probability proportional to their target density values. We also investigate a multivariate Gaussian-based symmetric multiproposal mechanism and prove that it also enjoys the same simplified acceptance step. This insight leads to significant theoretical and practical speedups. While both algorithms enjoy natural parallelizability, we show that conventional implementations are sufficient to confer efficiency gains across an array of dimensions and a number of target distributions.