Bayesian extensions of non-negative matrix factorization

Although non-negative matrix factorization has become a popular data analysis tool for non-negative data sets, there are still some issues remaining partly unsolved. We investigate the potential of Bayesian techniques towards the solution of two important open questions concerning uniqueness and actual number of sources underlying the data. We derive a general Bayesian optimality condition for NMF solutions and elaborate on the criterion for the Gaussian likelihood case. We further derive a variational Bayes NMF algorithm for the Gaussian likelihood using rectified Gaussian prior distributions and study its ability to estimate the true number of sources in a toy data set.