Investigating the performance of multi-objective optimization when learning Bayesian Networks

Bayesian Networks have been widely used in the last decades in many fields, to describe statistical dependencies among random variables. In general, learning the structure of such models is a problem with considerable theoretical interest that poses many challenges. On the one hand, it is a well-known NP-complete problem, practically hardened by the huge search space of possible solutions. On the other hand, the phenomenon of I-equivalence, i.e., different graphical structures underpinning the same set of statistical dependencies, may lead to multimodal fitness landscapes further hindering maximum likelihood approaches to solve the task. Despite all these difficulties, greedy search methods based on a likelihood score coupled with a regularizator score to account for model complexity, have been shown to be surprisingly effective in practice. In this paper, we consider the formulation of the task of learning the structure of Bayesian Networks as an optimization problem based on a likelihood score, without complexity terms to regularize it. In particular, we exploit the NSGA-II multi-objective optimization procedure in order to explicitly account for both the likelihood of a solution and the number of selected arcs, by setting these as the two objective functions of the method. The aim of this work is to investigate the behavior of NSGA-II and analyse the quality of its solutions. We thus thoroughly examined the optimization results obtained on a wide set of simulated data, by considering both the goodness of the inferred solutions in terms of the objective functions values achieved, and by comparing the retrieved structures with the ground truth, i.e., the networks used to generate the target data. Our results show that NSGA-II can converge to solutions characterized by better likelihood and less arcs than classic approaches, although paradoxically characterized in many cases by a lower similarity with the target network.