Branch and Bound for Regular Bayesian Network Structure Learing

We consider efficient Bayesian network structure learning (BNSL) based on scores using branch and bound. Thus far, as a BNSL score, the Bayesian Dirichlet equivalent uniform (BDeu) has been used most often, but it is recently proved that the BDeu does not choose the simplest model even when the likelihood is maximized whereas Jeffreys’ prior and MDL satisfy such regularity. Although the BDeu has been preferred because it gives Markov equivalent models the same score, in this paper, we introduce another class of scores (quotient scores) that satisfies the property, and propose a pruning rule for the quotient score based on Jeffreys’ prior. We find that the quotient score based on Jeffreys’ prior is regular, and that the proposed pruning rule utilizes the regularity, and is applied much more often than that of the BDeu, so that much less computation is required in BNSL. Finally, our experiments support the hypothesis that the regular scores outperform the non-regular ones in the sense of computational efficiency as well as correctness of BNSL.