Optimality of Multiple Decision Statistical Procedure for Gaussian Graphical Model Selection
Gaussian graphical model selection is a statistical problem
that identifies the Gaussian graphical model from observations. Existing
Gaussian graphical model selection methods focus on the error rate
for incorrect edge inclusion. However, when comparing statistical procedures,
it is also important to take into account the error rate for
incorrect edge exclusion. To handle this issue we consider the graphical
model selection problem in the framework of multiple decision theory.We
show that the statistical procedure based on simultaneous inference with
UMPU individual tests is optimal in the class of unbiased procedures.
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.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.