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Graphical models are increasingly being used in many complex engineering problems to model the dynamics between states of the graph. These graphs are often very large and approximation models are needed to reduce the computational complexity. This paper considers the problem of quantifying the quality of an approximation model for a graphical model (model selection problem). The model selection often uses a distance measure such as the Kullback–Leibler (KL) divergence between the original distribution and the model distribution to quantify the quality of the model approximation. We extend and broaden the body of research by formulating the model approximation as a detection problem between the original distribution and the model distribution. We focus on Gaussian random vectors and introduce the Correlation Approximation Matrix (CAM) and use the Area Under the Curve (AUC) for the formulated detection problem. The closeness measures such as the KL divergence, the log-likelihood ratio, and the AUC are functions of the eigenvalues of the CAM. Easily computable upper and lower bounds are found for the AUC. The paper concludes by computing these measures for real and synthetic simulation data. Tree approximations and more complex graphical models are considered for approximation models.