The Probabilistic Independent Component Analysis model presented in this paper is aimed at solving the problem of overfitting in classical ICA applied to FMRI data. This is approached by including a noise term in the classical ICA decomposition which renders the model identical to the standard GLM with the conceptual difference that the number and the shape of the regressors is estimated rather than pre-specified. As in the standard GLM, the noise is assumed to be additive and Gaussian distributed. Structured noise (e.g. physiological noise) is most likely to appear in the data as structured non-Gaussian noise, and as such is estimated as one (or more) of the underlying sources; this should not be confused with the Gaussian noise eliminated during the PPCA stage. If different source processes (e.g. activation and physiological noise components) are partially (temporally) correlated, they can still be separated from each other in the PICA unmixing as long as they are spatially distinct and not perfectly temporally correlated. If (as e.g. suggested by [Kruger and Glover, 2001]) a noise component combines non-additively with a signal source, then indeed the linear mixing model used here will be imperfect. In this case, however, the nonlinear interaction should (to first order) appear as a third PICA component, in exactly the same way as modelling nonlinear interactions as a third explanatory variable in GLM modelling attempts to do.
Some of the methodological steps presented in section 2 build on ideas and techniques from standard parametric FMRI modelling; for example, the estimation of the voxel-wise covariance for pre-whitening is an extension of the technique presented in [Woolrich et al., 2001]. Also, the use of mixture models for inference has been motivated by work from [Everitt and Bullmore, 1999] and [Hartvig and Jensen, 2000], where mixture models were used for statistical maps generated from parametric FMRI activation modelling and links to the work on explicit source density modelling for ICA [Attias, 1999,Choudrey and Roberts, 2001,Rowe, 2001].
The proposed methodology can be extended in various ways. In the present implementation, we chose to discard an explicit source model from the estimation stages and use the Gaussian mixture model only after estimation is completed for the inferential steps. In a more integrated approach, the mixture model could be re-estimated after every iteration. This could then be used as an alternative to neg-entropy estimation in order to explicitly quantify the non-Gaussianity of source processes. [Choudrey and Roberts, 2001] approximate full posterior distributions for all model parameters of a PICA model using the variational Bayesian framework. The technique is conceptually attractive, but suffers from a substantial increase in computational load and as such does not yet appear to be applicable to FMRI data. Also, our technique only encodes spatial neighbourhood information via the covariance of the observations that feeds into the PPCA step. In order to incorporate spatial information explicitly into the ICA estimation, a spatial Markov model can be used to represent the joint probability density of neighbouring samples.