Bayesian networks, and especially their structures, are powerful tools for representing conditional independencies and dependencies between random variables. In applications where related variables form a priori known groups, chosen to represent different "views" to or aspects of the same entities, one may be more interested in modeling dependencies between groups of variables rather than between individual variables. Motivated by this, we study prospects of representing relationships between variable groups using Bayesian network structures. We show that for dependency structures between groups to be learnable, the data have to satisfy the so-called groupwise faithfulness assumption. We also show that one cannot learn causal relations between groups using only groupwise conditional independencies, but also variable-wise relations are needed. Additionally, we present algorithms for finding the groupwise dependency structures.
Bayesian network structure identification is known to be NP-Hard in the general case. We demonstrate a heuristic search for structure identification based on aggregationhierarchies. The basic idea is to perform initial exhaustive searches on composite “high-level” random variables (RVs) that are created via aggregations of atomic RVs. The results of the high-level searches then constrain a refined search on the atomic RVs. We demonstrate our methods on a challenging real-world neuroimaging domain and show that they consistently yield higher scoring networks when compared to traditional searches, provided sufficient topological complexity is permitted. On simulated data, where ground truth is known and controllable, our methods yield improved classification accuracy and structural precision, but can also result in reduced structural recall on particularly noisy datasets.