The covariance selection quality for graphs with junction trees through AUC bounds

We conduct a study of graphical models and discuss the quality of model selection approximation by formulating the problem as a detection problem and examine the area under the curve (AUC). We are specifically looking at the model selection problem for jointly Gaussian random vectors. For Gaussian distributions, this problem simplifies to the covariance selection problem which is widely discussed in literature by Dempster [1]. In this paper, we discuss graphical models such as the pth order Markov chain and the pth order star network interpretation which also have junction tree graphical representations and give the definition for the correlation approximation matrix (CAM) which contains all information about the model selection problem. We compute the model covariance matrix as well as the KL divergence between the original distribution and the approximated model distribution. We conduct some simulations which show that the quality of the selected model increases as the model order, p, increases.