Dimension‐independent Markov chain Monte Carlo on the sphere

Abstract

AbstractWe consider Bayesian analysis on high‐dimensional spheres with angular central Gaussian priors. These priors model antipodally symmetric directional data, are easily defined in Hilbert spaces and occur, for instance, in Bayesian density estimation and binary level set inversion. In this paper we derive efficient Markov chain Monte Carlo methods for approximate sampling of posteriors with respect to these priors. Our approaches rely on lifting the sampling problem to the ambient Hilbert space and exploit existing dimension‐independent samplers in linear spaces. By a push‐forward Markov kernel construction we then obtain Markov chains on the sphere which inherit reversibility and spectral gap properties from samplers in linear spaces. Moreover, our proposed algorithms show dimension‐independent efficiency in numerical experiments.