The structure learning method used in BNW can be broken down into three main steps: calculating local network scores, determing global structures that optimize network scores, and, if indicated by user settings, performing model averaging over high scoring structures.
Local score calculation: The score of a local structure of a Bayesian network considers how well a node is explained by the nodes that are its immediate parents. To begin structure learning in BNW, we calculate all possible local scores by perform an exhaustive search of local structures given the structural constraints specified by the user. In order to allow structure learning of hybrid datasets, BNW calculates local scores using the scoring metric proposed by Bottcher and Dethlefsen that they have previously incorporated in the R package deal. Briefly, to allow for the use hybrid datasets containing both discrete and continuous variables, local structures are scored based on conditional probability tables for entirely discrete local strucutes, Gaussian distributions for entirely continuous local structures, and conditional Gaussian distributions for hybrid local structures. One property of this scoring metric is that it does not allow discrete nodes to be the children of continuous nodes. To improve the speed of structure learning, we do not use deal, but, instead, calculate local scores using code that we have written in the C programming language.
Search for the kbest global optimal structures: After calculating local scores, BNW performs a search for the kbest global optimal structures, using a userspecified value of k. The kbest structure search method used in BNW was developed by Tian and Re and is available here. It is an adaptation of an algorithm for identifying the optimal network structure developed by Silander and Myllymaki that is available here.
Model averaging: Model averaging can be used to reduce the risk of overfitting data to a single model. In BNW, model averaging is automatically performed over the kbest scoring structures, when users select values of k > 1. To select features (i.e., directed edges between nodes), the posterior probability of each feature is calculated by a weighted average over the kbest networks where the weight is given by the score of the global network structure. This posterior probability, which ranges for 1 for features included in all high scoring networks to 0 for features in no high scoring networks, reflects confidence in the feature.
Model averaging may be particularly advantageous when learning network models using small datasets. As the number of samples in a dataset increases, the differences between the scores of the highest scoring networks often also increase. Therefore, instead of a single network structure having the best score, structure learning of small datasets may identify a group of structures with similar scores. Model averaging can be used to select features that are common to many of these high scoring networks.
