  Recent reports have described that the equivalent sample size (ESS) in a
Dirichlet prior plays an important role in learning Bayesian networks. This
paper provides an asymptotic analysis of the marginal likelihood score for a
Bayesian network. Results show that the ratio of the ESS and sample size
determine the penalty of adding arcs in learning Bayesian networks. The number
of arcs increases monotonically as the ESS increases; the number of arcs
monotonically decreases as the ESS decreases. Furthermore, the marginal
likelihood score provides a unified expression of various score metrics by
changing prior knowledge.
