A comparison of score-based methods for estimating Bayesian networks using the Kullback-Leibler divergence

In this paper, we compare the performance of two methods for estimating Bayesian networks from data containing exogenous variables and random effects. The first method is fully Bayesian in which a prior distribution is placed on the exogenous variables, whereas the second method, which we call the residual approach, accounts for the effects of exogenous variables by using the notion of restricted maximum likelihood. We review the two score-based metrics, then study their performance by measuring the Kullback Leibler divergence, or distance, between the two resulting posterior density functions. The Kullback Leibler divergence provides a natural framework for comparing distributions. The residual approach is considerably simpler to apply in practice and we demonstrate its utility both theoretically and via simulations. In particular, in applications where the exogenous variables are not of primary interest, we show that the potential loss of information about parameters and induced components of correlation, is generally small.