Accurate directional inference in Gaussian graphical models

Directional tests to compare incomplete undirected graphs are developed in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation leads to exceptional accuracy of the proposed approach. This is verified by simulation experiments with high-dimensional parameters of interest, where inference via standard asymptotic approximations to the likelihood ratio test and some of its higher-order modifications fails. The directional $p$-value is used to illustrate the assessment of Markovian dependencies in a dataset from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.