Loss function, unbiasedness, and optimality of Gaussian graphical model selection

Abstract

A Gaussian graphical model is a graphical representation of the dependence structure for a Gaussian random vector. Gaussian graphical model selection is a statistical problem that identifies the Gaussian graphical model from observations. There are several statistical approaches for Gaussian graphical model identification. Their properties, such as unbiasedeness and optimality, are not established. In this paper we study these properties. We consider the graphical model selection problem in the framework of multiple decision theory and suggest assessing these procedures using an additive loss function. Associated risk function in this case is a linear combination of the expected numbers of the two types of error (False Positive and False Negative). We combine the tests of a Neyman structure for individual hypotheses with simultaneous inference and prove that the obtained multiple decision procedure is optimal in the class of unbiased multiple decision procedures.