  We conduct a study of graphical models and discuss the quality of model
selection approximation by formulating the problem as a detection problem and
examining the area under the curve (AUC). We are specifically looking at the
model selection problem for jointly Gaussian random vectors. For Gaussian
random vectors, this problem simplifies to the covariance selection problem
which is widely discussed in literature by Dempster [1]. In this paper, we give
the definition for the correlation approximation matrix (CAM) which contains
all information about the model selection problem and discuss the pth order
Markov chain model and the $p$th order star network model for the a Gaussian
distribution with Toeplitz covariance matrix. For each model, we compute the
model covariance matrix as well as the KL divergence between the Gaussian
distribution and its model. We also show that if the model order, p, is
proportional to the number of nodes, n, then the model selection is
asymptotically good as the number of nodes, n, goes to infinity since the AUC
in this case is bounded away from one. We conduct some simulations which
confirm the theoretical analysis and also show that the selected model quality
increases as the model order, p, increases.
