Unsupervised learning for source separation with mixture of Gaussians prior for sources and Gaussian prior for mixture coefficients

The authors present two new algorithms for unsupervised learning and source separation for the case of noisy instantaneous linear mixture, within the Bayesian inference framework. The source distribution prior is modeled by a mixture of Gaussians (E. Moulines, 1997) and the mixing matrix elements distributions by a Gaussian. We model the mixture of Gaussians hierarchically by means of hidden variables representing the labels of the mixture. Then, we consider the joint a posteriori distribution of sources, mixing matrix elements, labels of the mixture and other parameters of the mixture with appropriate prior probability laws to eliminate degeneracy of the likelihood function of variance parameters. We also propose two algorithms to estimate sources, mixing matrix and hyperparameters: Joint MAP (maximum a posteriori) algorithm and penalized EM-type algorithm. The performances of these two algorithms are compared through an illustrative example taken by O. Macchi and E. Moreau (1999).