Variational Nonnegative Matrix Factorisation

We describe non-negative matrix factorisation (NMF) in a statistical framework, with a hierarchical generative model consisting of an observation and a prior component. Omitting the prior leads to standard NMF algorithms as special cases, where maximum likelihood parameter estimation is carried out via the expectation-maximisation (EM) algorithm. Starting from this view, we develop Bayesian extensions that facilitate more powerful modelling and allow more sophisticated inference, such as Bayesian model selection. Our construction retains conjugacy and enables us to develop models that fit better to real data while retaining attractive features of standard NMF such as fast convergence and easy implementation. We illustrate our approach on model order selection and image reconstruction.