Comment on Geodesic Monte Carlo on Embedded Manifolds by Byrne and Girolami

摘要

The authors have introduced an interesting and mathematically intricate method for Markov chain Monte Carlo (MC) simulation on an embedded manifold. The geodesic MC method provides large proposals as part of the scheme, which are devised by careful study of the Riemannian geometry of the space and the geodesies in particular. The aim of the resulting algorithm is to produce a chain with low autocorrelation and high acceptance probabilities. As displayed by the authors, the method is well geared up for simulating from unimodal distributions on a manifold via the gradient of the log-density and the geodesic flow. They also demonstrate its effective use in multimodal scenarios via parallel tempering. Given that there are always many choices of embedding, should one choose as low-dimensional embedding as possible?