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We consider the problem of quantifying the quality of a model selectionproblem for a graphical model. We discuss this by formulating the problem as adetection problem. Model selection problems usually minimize a distance betweenthe original distribution and the model distribution. For the special case ofGaussian distributions, the model selection problem simplifies to thecovariance selection problem which is widely discussed in literature byDempster [2] where the likelihood criterion is maximized or equivalently theKullback-Leibler (KL) divergence is minimized to compute the model covariancematrix. While this solution is optimal for Gaussian distributions in the senseof the KL divergence, it is not optimal when compared with other informationdivergences and criteria such as Area Under the Curve (AUC).  In this paper, we analytically compute upper and lower bounds for the AUC anddiscuss the quality of model selection problem using the AUC and its bounds asan accuracy measure in detection problem. We define the correlationapproximation matrix (CAM) and show that analytical computation of the KLdivergence, the AUC and its bounds only depend on the eigenvalues of CAM. Wealso show the relationship between the AUC, the KL divergence and the ROC curveby optimizing with respect to the ROC curve. In the examples provided, we picktree structures as the simplest graphical models. We perform simulations onfully-connected graphs and compute the tree structured models by applying thewidely used Chow-Liu algorithm [3]. Examples show that the quality of treeapproximation models are not good in general based on information divergences,the AUC and its bounds when the number of nodes in the graphical model islarge. We show both analytically and by simulations that the 1-AUC for the treeapproximation model decays exponentially as the dimension of graphical modelincreases.

The Quality of the Covariance Selection Through Detection Problem and AUC Bounds