  In this paper, we first consider the parameter estimation of a multivariate
random process distribution using multivariate Gaussian mixture law. The labels
of the mixture are allowed to have a general probability law which gives the
possibility to modelize a temporal structure of the process under study. We
generalize the case of univariate Gaussian mixture in [Ridolfi99] to show that
the likelihood is unbounded and goes to infinity when one of the covariance
matrices approaches the boundary of singularity of the non negative definite
matrices set. We characterize the parameter set of these singularities. As a
solution to this degeneracy problem, we show that the penalization of the
likelihood by an Inverse Wishart prior on covariance matrices results to a
penalized or maximum a posteriori criterion which is bounded. Then, the
existence of positive definite matrices optimizing this criterion can be
guaranteed. We also show that with a modified EM procedure or with a Bayesian
sampling scheme, we can constrain covariance matrices to belong to a particular
subclass of covariance matrices. Finally, we study degeneracies in the source
separation problem where the characterization of parameter singularity set is
more complex. We show, however, that Inverse Wishart prior on covariance
matrices eliminates the degeneracies in this case too.
