  Score functions for learning the structure of Bayesian networks in the
literature assume that data are a homogeneous set of observations; whereas it
is often the case that they comprise different related, but not homogeneous,
data sets collected in different ways. In this paper we propose a new Bayesian
Dirichlet score, which we call Bayesian Hierarchical Dirichlet (BHD). The
proposed score is based on a hierarchical model that pools information across
data sets to learn a single encompassing network structure, while taking into
account the differences in their probabilistic structures. We derive a
closed-form expression for BHD using a variational approximation of the
marginal likelihood, we study the associated computational cost and we evaluate
its performance using simulated data. We find that, when data comprise multiple
related data sets, BHD outperforms the Bayesian Dirichlet equivalent uniform
(BDeu) score in terms of reconstruction accuracy as measured by the Structural
Hamming distance, and that it is as accurate as BDeu when data are homogeneous.
This improvement is particularly clear when either the number of variables in
the network or the number of observations is large. Moreover, the estimated
networks are sparser and therefore more interpretable than those obtained with
BDeu thanks to a lower number of false positive arcs.
