Matrix factorisation treats observations as linear combinations of basis vectors together with, possibly, additive noise. Notable techniques in this family are Principal Components Analysis and Independent Components Analysis. Applied to brain images, matrix factorisation provides insight into the spatial and temporal structure of data. We improve on current practice with methods that unify different stages of analysis simultaneously for all subjects in a dataset, including dimension estimation and reduction. This results in uncertainty information being carried coherently through the analysis. A computationally efficient approach to correlated multivariate normal distributions is set out. This enables spatial smoothing during the inference of basis vectors, to a level determined by the data. Applied to neuroimaging, this reduces the need for blurring of the data during preprocessing. Orthogonality constraints on the basis are relaxed, allowing for overlapping ‘networks’ of activity. We consider a nonparametric matrix factorisation model inferred using Markov Chain Monte Carlo (MCMC). This approach incorporates dimensionality estimation into the infer- ence process. Novel parallelisation strategies for MCMC on repeated graphs are provided to expedite inference. In simulations, modelling correlation structure is seen to improve source separation where latent basis vectors are not orthogonal. The Cambridge Centre for Ageing and Neuroscience (Cam-CAN) project obtained fMRI data while subjects watched a short film, on 30 of whose recordings we demonstrate the approach. To conduct inference on larger datasets, we provide a fixed dimension Structured Matrix Factorisation (SMF) model, inferred through Variational Bayes (VB). By modelling the components as a mixture, more general distributions can be expressed. The VB approach scaled to 600 subjects from Cam-CAN, enabling a comparison to, and validation of, the main findings of an earlier analysis; notably that subjects’ responses to movie watching became less synchronised with age. We discuss differences in results obtained under the MCMC and VB inferred models.