  Learning the structure of dependence relations between variables is a
pervasive issue in the statistical literature. A directed acyclic graph (DAG)
can represent a set of conditional independences, but different DAGs may encode
the same set of relations and are indistinguishable using observational data.
Equivalent DAGs can be collected into classes, each represented by a partially
directed graph known as essential graph (EG). Structure learning directly
conducted on the EG space, rather than on the allied space of DAGs, leads to
theoretical and computational benefits. Still, the majority of efforts in the
literature has been dedicated to Gaussian data, with less attention to methods
designed for multivariate categorical data. We then propose a Bayesian
methodology for structure learning of categorical EGs. Combining a constructive
parameter prior elicitation with a graph-driven likelihood decomposition, we
derive a closed-form expression for the marginal likelihood of a categorical EG
model. Asymptotic properties are studied, and an MCMC sampler scheme developed
for approximate posterior inference. We evaluate our methodology on both
simulated scenarios and real data, with appreciable performance in comparison
with state-of-the-art methods.
