Learning Bayesian networks. II. A computational algorithm

For pt. I see ibid., pp. 769-76. Following the theory developed in a companion paper, this paper presents a general algorithm for learning Bayesian networks from data. The algorithm implements a global optimisation of a joint MAP-MDL score aiming at discovering the best Bayesian network model underlying the given data set. The whole procedure is divided into three phases. The first phase aims at finding an undirected structure - a possibly minimal potential graph - which approximately minimizes the description length of the DAG (directed acyclic graph) structure. This is realized through a potential graph (PG) algorithm which successively prunes the fully connected potential graph based on conditional independence tests. The second phase aims at extracting a DAG structure from the. minimal potential graph obtained from the first phase, using an axiomatic and an inductive causality discovery (ACD, ICD) algorithm. The ACD algorithm determines the directions of a subset of undirected links in the potential graph using axiomatic graph-theoretical constraints on d-separations. The ICD algorithm determines the direction of remaining undirected links through a global optimisation under the joint criterion. The structural search space is substantially reduced by the subset of directed links determined by the ACD algorithm and the subset of undirected links obtained by the PG algorithm. Finally at the third phase, to correct possible errors on link omission or commission, the structure refinement (SR) algorithm evaluates the loss of each cut link and the worthiness of each determined link still using the joint MAP-MDL criterion.