An empirical Bayes approach for joint Bayesian model selection and estimation of sinusoids via reversible jump MCMC

This paper addresses the sensitivity of the algorithm proposed by Andrieu and Doucet (IEEE Trans. Signal Process., 47(10), 1999), for the joint Bayesian model selection and estimation of sinusoids in white Gaussian noise, to the values of a certain hyperparameter claimed to be weakly influential in the original paper. A deeper study of this issue reveals indeed that the value of this hyperparameter (the scale parameter of the expected signal-to-noise ratio) has a significant influence on 1) the mixing rate of the Markov chain and 2) the posterior distribution of the number of components. As a possible workaround for this problem, we investigate an Empirical Bayes approach to select an appropriate value for this hyperparameter in a data-driven way. Marginal likelihood maximization is performed by means of an importance sampling based Monte Carlo EM (MCEM) algorithm. Numerical experiments illustrate that the sampler equipped with this MCEM procedure provides satisfactory performances in moderate to high SNR situations.