  Analyzing multiple studies allows leveraging data from a range of sources and
populations, but until recently, there have been limited methodologies to
approach the joint unsupervised analysis of multiple high-dimensional studies.
A recent method, Bayesian Multi-Study Factor Analysis (BMSFA), identifies
latent factors common to all studies, as well as latent factors specific to
individual studies. However, BMSFA does not allow for partially shared factors,
i.e. latent factors shared by more than one but less than all studies. We
extend BMSFA by introducing a new method, Tetris, for Bayesian combinatorial
multi-study factor analysis, which identifies latent factors that can be shared
by any combination of studies. We model the subsets of studies that share
latent factors with an Indian Buffet Process. We test our method with an
extensive range of simulations, and showcase its utility not only in dimension
reduction but also in covariance estimation. Finally, we apply Tetris to
high-dimensional gene expression datasets to identify patterns in breast cancer
gene expression, both within and across known classes defined by germline
mutations.
