  The problem of source separation is by its very nature an inductive inference
problem. There is not enough information to deduce the solution, so one must
use any available information to infer the most probable solution. We
demonstrate that source separation problems are well-suited for the Bayesian
approach which provides a natural and logically consistent method by which one
can incorporate prior knowledge to estimate the most probable solution given
that knowledge.
  We derive the Bell-Sejnowski ICA algorithm from first principles, i.e. Bayes'
Theorem and demonstrate how the Bayesian methodology makes explicit the
underlying assumptions. We then further demonstrate the power of the Bayesian
approach by deriving two separation algorithms that incorporate additional
prior information. One algorithm separates signals that are known a priori to
be decorrelated and the other utilizes information about the signal propagation
through the medium from the sources to the detectors.
