  In the Bayesian approach to structure learning of graphical models, the
equivalent sample size (ESS) in the Dirichlet prior over the model parameters
was recently shown to have an important effect on the maximum-a-posteriori
estimate of the Bayesian network structure. In our first contribution, we
theoretically analyze the case of large ESS-values, which complements previous
work: among other results, we find that the presence of an edge in a Bayesian
network is favoured over its absence even if both the Dirichlet prior and the
data imply independence, as long as the conditional empirical distribution is
notably different from uniform. In our second contribution, we focus on
realistic ESS-values, and provide an analytical approximation to the "optimal"
ESS-value in a predictive sense (its accuracy is also validated
experimentally): this approximation provides an understanding as to which
properties of the data have the main effect determining the "optimal"
ESS-value.
