Nonnegative matrix factorizations as probabilistic inference in composite models

We develop an interpretation of nonnegative matrix factorization (NMF) methods based on Euclidean distance, Kullback-Leibler and Itakura-Saito divergences in a probabilistic framework. We describe how these factorizations are implicit in a well-defined statistical model of superimposed components, either Gaussian or Poisson distributed, and are equivalent to maximum likelihood estimation of either mean, variance or intensity parameters. By treating the components as hidden-variables, NMF algorithms can be derived in a typical data augmentation setting. This setting can in particular accommodate regularization constraints on the matrix factors through Bayesian priors. We describe multiplicative, Expectation-Maximization, Markov chain Monte Carlo and Variational Bayes algorithms for the NMF problem. This paper describes in a unified framework both new and known algorithms and aims at providing statistical insights to NMF.