  Non-negative matrix factorization (NMF) has become a well-established class
of methods for the analysis of non-negative data. In particular, a lot of
effort has been devoted to probabilistic NMF, namely estimation or inference
tasks in probabilistic models describing the data, based for example on Poisson
or exponential likelihoods. When dealing with time series data, several works
have proposed to model the evolution of the activation coefficients as a
non-negative Markov chain, most of the time in relation with the Gamma
distribution, giving rise to so-called temporal NMF models. In this paper, we
review four Gamma Markov chains of the NMF literature, and show that they all
share the same drawback: the absence of a well-defined stationary distribution.
We then introduce a fifth process, an overlooked model of the time series
literature named BGAR(1), which overcomes this limitation. These temporal NMF
models are then compared in a MAP framework on a prediction task, in the
context of the Poisson likelihood.
