Confidence graphs for graphical model selection

In this article, we introduce the concept of confidence graphs (CG) for graphical model selection. CG first identifies two nested graphical models—called small and large confidence graphs (SCG and LCG)—trapping the true graphical model in between at a given level of confidence, just like the endpoints of traditional confidence interval capturing the population parameter. Therefore, SCG and LCG provide us with more insights about the simplest and most complex forms of dependence structure the true model can possibly be, and their difference also offers us a measure of model selection uncertainty. In addition, rather than relying on a single selected model, CG consists of a group of graphical models between SCG and LCG as the candidates. The proposed method can be coupled with many popular model selection methods, making it an ideal tool for comparing model selection uncertainty as well as measuring reproducibility. We also propose a new residual bootstrap procedure for graphical model settings to approximate the sampling distribution of the selected models and to obtain CG. To visualize the distribution of selected models and its associated uncertainty, we further develop new graphical tools, such as grouped model selection distribution plot. Numerical studies further illustrate the advantages of the proposed method.