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Bayesian inference methods for sources separation

The main aim of this paper is first to present the Bayesian inference approach for sources separation where we want to infer on the mixing matrix, the sources and all the hyperparameters associated to probabilistic modeling (likelihood and priors). For this purpose, the sources separation problem is considered in four steps: i) Estimation of the sources when the mixing matrix is known; ii) Estimation of the mixing matrix when the sources are known; iii) Joint estimation of sources and the mixing matrix; and finally, iv) Joint estimation of sources and the mixing matrix, hidden variables and hyper-parameters. In all cases, one of the main steps is modeling of sources and the mixing matrix prior laws. We propose to use sparsity enforcing probability laws (such as Generalized Gaussian, Student-t and mixture models) both for the sources and the mixing matrix. For algorithmic and computational aspects, we consider either Joint MAP, MCMC Gibbs sampling or Variational Bayesian Approximation tools. For each class of methods we discuss about their relative costs and performances.