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Reversible jump MCMC (RJ-MCMC) sampling techniques, which allow to jointlytackle model selection and parameter estimation problems in a coherent Bayesianframework, have become increasingly popular in the signal processing literaturesince the seminal paper of Andrieu and Doucet (IEEE Trans. Signal Process.,47(10), 1999). Crucial to the implementation of any RJ-MCMC sampler is thecomputation of the so-called Metropolis-Hastings-Green (MHG) ratio, whichdetermines the acceptance probability for the proposed moves.  It turns out that the expression of the MHG ratio that was given in the paperof Andrieu and Doucet for "Birth-or-Death" moves---the simplest kind oftrans-dimensional move, used in virtually all applications of RJ-MCMC to signaldecomposition problems---was erroneous. Unfortunately, this mistake has beenreproduced in many subsequent papers dealing with RJ-MCMC sampling in thesignal processing literature.  This note discusses the computation of the MHG ratio, with a focus on thecase where the proposal kernel can be decomposed as a mixture of simplerkernels, for which the MHG ratio is easy to compute. We provide sufficientconditions under which the MHG ratio of the mixture can be deduced from the MHGratios of the elementary kernels of which it is composed. As an application, weconsider the case of Birth-or-Death moves, and provide a corrected expressionfor the erroneous ratio in the paper of Andrieu and Doucet.

Note on the computation of the Metropolis-Hastings ratio for  Birth-or-Death moves in trans-dimensional MCMC algorithms for signal  decomposition problems