Generalizing Bayesian networks, maximal ancestral graphs (MAGs) are a theoretically appealing model class for dealing with unobserved variables. Despite significant advances in developing practical exact algorithms for learning score-optimal Bayesian networks, practical exact algorithms for learning score-optimal MAGs have not been developed to-date. We develop here methodology for score-based structure learning of directed maximal ancestral graphs. In particular, we develop local score computation employing a linear Gaussian BIC score, as well as score pruning techniques, which are essential for exact structure learning approaches. Furthermore, employing dynamic programming and branch and bound, we present a first exact search algorithm that is guaranteed to find a globally optimal MAG for given local scores. The experiments show that our approach is able to find considerably higher scoring MAGs than previously proposed in-exact approaches.

Peer reviewed

We present a new algorithmic approach for the task of finding a chordal Markov network structure that maximizes a given scoring function. The algorithm is based on branch and bound and integrates dynamic programming for both domain pruning and for obtaining strong bounds for search-space pruning. Empirically, we show that the approach dominates in terms of running times a recent integer programming approach (and thereby also a recent constraint optimization approach) for the problem. Furthermore, our algorithm scales at times further with respect to the number of variables than a state-of-the-art dynamic programming algorithm for the problem, with the potential of reaching 20 variables and at the same time circumventing the tight exponential lower bounds on memory consumption of the pure dynamic programming approach.