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Myopic reliance on a misleading first sentence in the abstract of Covariance Selectiona Dempster (1972) spawned the computationally and mathematically dysfunctional Gaussian graphical model (GGM). In stark contrast to the GGM approach, the actual (Dempster, 1972, § 3) algorithm facilitated elegant and powerful applications, including a “texture model” developed two decades ago involving arbitrary distributions of 1000+ dimensions Zhu (1996). The “Covariance Selection” algorithm proposes a greedy sequence of increasingly constrained maximum entropy hypotheses Good (1963), terminating when the observed data “fails to reject” the last proposed probability distribution. We are mathematically critical of GGM methods that address a continuous convex domain with a discrete domain “golden hammer”. Computationally, selection of the wrong tool morphs polynomial-time algorithms into exponential-time algorithms. GGMs concepts are at odds with the fundamental concept of the invariant spherical multivariate Gaussian distribution. We are critical of the Bayesian GGM approach because the model selection process derails at the start when virtually all prior mass is attributed to comically precise multi-dimensional geometric “configurations” (Dempster, 1969, Ch. 13). We propose two Bayesian alternatives. The first alternative is based upon (Dempster, 1969, Ch. 15.3) and (Hoff, 2009, Ch. 7). The second alternative is based upon Bretthorst (2012), a recent paper placing maximum entropy methods such as the “Covariance Selection” algorithm in a Bayesian framework.

The Wrong Tool for Inference - A Critical View of Gaussian Graphical Models