  We introduce a factor analysis model that summarizes the dependencies between
observed variable groups, instead of dependencies between individual variables
as standard factor analysis does. A group may correspond to one view of the
same set of objects, one of many data sets tied by co-occurrence, or a set of
alternative variables collected from statistics tables to measure one property
of interest. We show that by assuming group-wise sparse factors, active in a
subset of the sets, the variation can be decomposed into factors explaining
relationships between the sets and factors explaining away set-specific
variation. We formulate the assumptions in a Bayesian model which provides the
factors, and apply the model to two data analysis tasks, in neuroimaging and
chemical systems biology.
