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Computational Intelligence and Neuroscience
Volume 2009, Article ID 785152, 17 pages
Research Article

Bayesian Inference for Nonnegative Matrix Factorisation Models

Department of Computer Engineering, Boğaziçi University, 34342 Bebek, Istanbul, Turkey

Received 29 August 2008; Accepted 14 February 2009

Academic Editor: S. Cruces-Alvarez

Copyright © 2009 Ali Taylan Cemgil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We describe nonnegative matrix factorisation (NMF) with a Kullback-Leibler (KL) error measure in a statistical framework, with a hierarchical generative model consisting of an observation and a prior component. Omitting the prior leads to the standard KL-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 full Bayesian inference via variational Bayes or Monte Carlo. Our construction retains conjugacy and enables us to develop more powerful models while retaining attractive features of standard NMF such as monotonic convergence and easy implementation. We illustrate our approach on model order selection and image reconstruction.