  Theory of graphical models has matured over more than three decades to
provide the backbone for several classes of models that are used in a myriad of
applications such as genetic mapping of diseases, credit risk evaluation,
reliability and computer security, etc. Despite of their generic applicability
and wide adoptance, the constraints imposed by undirected graphical models and
Bayesian networks have also been recognized to be unnecessarily stringent under
certain circumstances. This observation has led to the proposal of several
generalizations that aim at more relaxed constraints by which the models can
impose local or context-specific dependence structures. Here we consider an
additional class of such models, termed as stratified graphical models. We
develop a method for Bayesian learning of these models by deriving an
analytical expression for the marginal likelihood of data under a specific
subclass of decomposable stratified models. A non-reversible Markov chain Monte
Carlo approach is further used to identify models that are highly supported by
the posterior distribution over the model space. Our method is illustrated and
compared with ordinary graphical models through application to several real and
synthetic datasets.
