On learning the structure of Bayesian Networks and submodular function maximization

Learning the structure of dependencies among multiple random variables is a problem of considerable theoretical and practical interest. In practice, score optimisation with multiple restarts provides a practical and surprisingly successful solution, yet the conditions under which this may be a well founded strategy are poorly understood. In this paper, we prove that the problem of identifying the structure of a Bayesian Network via regularised score optimisation can be recast, in expectation, as a submodular optimisation problem, thus guaranteeing optimality with high probability. This result both explains the practical success of optimisation heuristics, and suggests a way to improve on such algorithms by artificially simulating multiple data sets via a bootstrap procedure. We show on several synthetic data sets that the resulting algorithm yields better recovery performance than the state of the art, and illustrate in a real cancer genomic study how such an approach can lead to valuable practical insights.