Flexible and efficient implementations of Bayesian independent component analysis

In this paper we present an empirical Bayes method for flexible and efficient independent component analysis (ICA). The method is flexible with respect to choice of source prior, dimensionality and constraints of the mixing matrix (unconstrained or non-negativity), and structure of the noise covariance matrix. Parameter optimization is handled by variants of the expectation maximization (EM) algorithm: overrelaxed adaptive EM and the easy gradient recipe. These retain the simplicity of EM while converging faster. The required expectations over the source posterior, the sufficient statistics, are estimated with mean field methods: variational and the expectation consistent (EC) framework. We describe the derivation of the EC framework for ICA in detail and give empirical results demonstrating the improved performance. The paper is accompanied by the publicly available Matlab toolbox icaMF.