Bayesian Source Separation

Solving a Source separation problem using a maximum likelihood approach offers the possibility to encode, in addition to the mutual statistical independence of the sources, additional prior information on these signals by specifying their probability distribution functions. Such setting also corresponds to some specific choice of the non-linear functions in independent component analysis (ICA) algorithms based on non-linear decorrelation (See chapter \ref{chap:1} of this book). The Bayesian inference strategy offers an additional flexibility by allowing to take into account the noise statistics and to account for prior information on the mixing coefficients. The purpose of this chapter is to present the Bayesian approach for source separation. The general framework of Bayesian estimation will be presented in the first section of this chapter. It includes the specification of the likelihood resulting from the statistical description of the noise and the formulation of statistical models encoding the available information on the sought source signals and mixing coefficients. Resulting algorithms in the case of linear or nonlinear mixing models in the context of physical-chemical sensing will be presented. The Bayesian approach will be illustrated through some examples of case studies based on spectral data resulting from spectrometry measurements.