Introduction

In this chapter we investigate ICA models in which the number of sources, M, may be less than the number of sensors, N: so-called non-square mixing.

The ‘extra’ sensor observations are explained as observation noise. This general approach may be called Probabilistic Independent Component Analysis (PICA) by analogy with the Probabilistic Principal Component Analysis (PPCA) model of Tipping & Bishop [I9971; ICA and PCA don't have observation noise, PICA and PPCA do.

Non-square ICA models give rise to a likelihood model for the data involving an integral which is intractable. In this chapter we build on previous work in which the integral is estimated using a Laplace approximation. By making the further assumption that the unmixing matrix lies on the decorrelating manifold we are able to make a number of simplifications. Firstly, the observation noise can be estimated using PCA methods, and, secondly, optimisation takes place in a space having a much reduced dimensionality, having order M2 parameters rather than M × N. Again, building on previous work, we derive a model order selection criterion for selecting the appropriate number of sources. This is based on the Laplace approximation as applied to the decorrelating manifold. This is then compared with PCA model order selection methods on music and EEG datasets.