  We present novel understandings of the Gamma-Poisson (GaP) model, a
probabilistic matrix factorization model for count data. We show that GaP can
be rewritten free of the score/activation matrix. This gives us new insights
about the estimation of the topic/dictionary matrix by maximum marginal
likelihood estimation. In particular, this explains the robustness of this
estimator to over-specified values of the factorization rank, especially its
ability to automatically prune irrelevant dictionary columns, as empirically
observed in previous work. The marginalization of the activation matrix leads
in turn to a new Monte Carlo Expectation-Maximization algorithm with favorable
properties.
