Mikkel N. Schmidt, Ole Winther, and Lars Kai Hansen

Abstract: We present a Bayesian treatment of non-negative matrix factorization (NMF), based on a normal likelihood and exponential priors, and derive an efficient Gibbs sampler to approximate the posterior density of the NMF factors. On a chemical brain imaging data set, we show that this improves interpretability by providing uncertainty estimates. We discuss how the Gibbs sampler can be used for model order selection by estimating the marginal likelihood, and compare with the Bayesian information criterion. For computing the maximum a posteriori estimate we present an iterated conditional modes algorithm that rivals existing state-of-the-art NMF algorithms on an image feature extraction problem.

Application example: Analysis of chemical shift imaging data. Two components are identified corresponding to (top) muscle and (bottom) brain tissue. MAP estimation provides a point estimate of the components whereas Gibbs sampling gives full posterior marginals, that provide an uncertainty estimate on the spectra, and leads to better interpretation of the results. For example, the confidence intervals show, that many of the low amplitude peaks in the MAP spectra may not be significant.

Conclusions: We have taken a Bayesian approach to NMF and presented a fast MCMC sampling procedure for approximating the posterior density, and we have showed that this can be valuable for the interpretation of the non-negative factors recovered in NMF. The sampling procedure can also directly be used to estimate the marginal likelihood, which is useful for model order selection. Finally, we have presented an iterated conditional modes algorithm for computing the MAP estimate, that rivals existing state-of-the-art NMF algorithms.