Independent Component Analysis is becoming a popular exploratory method for analysing complex data such as that from FMRI experiments. The application of such `model-free' methods, however, has been somewhat restricted both by the view that results can be uninterpretable and by the lack of ability to quantify statistical significance. We present an integrated approach to Probabilistic ICA for FMRI data that allows for non-square mixing in the presence of Gaussian noise. We employ an objective estimation of the amount of Gaussian noise through Bayesian analysis of the true dimensionality of the data, i.e. the number of activation and non-Gaussian noise sources. Reduction of the data to this `true' subspace before the ICA decomposition automatically results in an estimate of the noise, leading to the ability to assign significance to voxels in ICA spatial maps. By this we not only are able to carry out probabilistic modelling, but also reduce problems of interpretation and ove...