Using complexity for the estimation of Bayesian networks

Statistical inference of graphical models has become an important tool in the reconstruction of biological networks of the type which model, for example, gene regulatory interactions. In particular, the construction of a score-based Bayesian posterior density over the space of models provides an intuitive and computationally feasible method of assessing model uncertainty and of assigning statistical confidence to structural features. One problem which frequently occurs with this approach is the tendency to overestimate the degree of model complexity. Spurious graphical features obtained in this way may affect the inference in unpredictable ways, even when using scoring techniques, such as the Bayesian Information Criterion (BIC), that are specifically designed to compensate for overfitting. In this article we propose a simple adjustment to a BIC-based scoring procedure. The method proceeds in two steps. In the first step we derive an independent estimate of the parametric complexity of the model. In the second we modify the BIC score so that the mean parametric complexity of the posterior density is equal to the estimated value. The method is applied to a set of test networks, and to a collection of genes from the yeast genome known to possess regulatory relationships. A Bayesian network model with binary responses is employed. In the examples considered, we find that the number of spurious graph edges inferred is reduced, while the effect on the identification of true edges is minimal.