Note on the computation of the Metropolis-Hastings ratio for   Birth-or-Death moves in trans-dimensional MCMC algorithms for signal   decomposition problems
  Reversible jump MCMC (RJ-MCMC) sampling techniques, which allow to jointly tackle model selection and parameter estimation problems in a coherent Bayesian framework, have become increasingly popular in the signal processing literature since the seminal paper of Andrieu and Doucet (IEEE Trans. Signal Process., 47(10), 1999). Crucial to the implementation of any RJ-MCMC sampler is the computation of the so-called Metropolis-Hastings-Green (MHG) ratio, which determines the acceptance probability for the proposed moves.   It turns out that the expression of the MHG ratio that was given in the paper of Andrieu and Doucet for "Birth-or-Death" moves---the simplest kind of trans-dimensional move, used in virtually all applications of RJ-MCMC to signal decomposition problems---was erroneous. Unfortunately, this mistake has been reproduced in many subsequent papers dealing with RJ-MCMC sampling in the signal processing literature.   This note discusses the computation of the MHG ratio, with a focus on the case where the proposal kernel can be decomposed as a mixture of simpler kernels, for which the MHG ratio is easy to compute. We provide sufficient conditions under which the MHG ratio of the mixture can be deduced from the MHG ratios of the elementary kernels of which it is composed. As an application, we consider the case of Birth-or-Death moves, and provide a corrected expression for the erroneous ratio in the paper of Andrieu and Doucet. 
Alireza Roodaki and Julien Bect and Gilles Fleury
stat.CO