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

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

2011-11-27

arxiv.org/abs/1111.6245v1

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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. This note discusses the computation of the MHG ratio, in 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---the simplest kind of trans-dimensional move, which is used in virtually all applications of RJ-MCMC to signal decomposition problems. It turns out that the expression of the MHG ratio that was given in the paper of Andrieu and Doucet was erroneous. Unfortunately, this mistake has contaminated most subsequent papers dealing with RJ-MCMC sampling in the signal processing literature.

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