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

Journal of Statistical Planning and Inference. 2019. Vol. 201. P. 32–39.

Kalyagin V. A., Koldanov A. P., Koldanov P., Pardalos P. M.

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.

Keywords: Loss function, Risk function, Gaussian graphical model, Optimal multiple decision statistical procedures, Unbiased multiple decision statistical procedures, Tests of a Neyman structure

Publication based on the results of: Methods and Algorithms for Networks Analysis (2019)