Our ability to extract the maximal amount of information from future observations at gigahertz frequencies depends on our ability to separate the underlying cosmic microwave background (CMB) from galactic and extragalactic foregrounds. We review the separation problem and its formulation within Bayesian inference, give examples of specific solutions with particular choices of prior density, and finally comment on the generalization of Bayesian methods to a multi-resolution framework. We propose

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... a strategy for the regularization of solutions allowing a spatially varying spectral index, and discuss possible computational approaches such as multi-scale stochastic relaxation.