Some properties of the expectation coordinate system of a binary exponential (distribution) family

Abstract

It has been shown that there exist monotonic relations between the elements of the normalized Fisher information matrix (correlation coefficients) and the natural parameters, if a certain inclusion relation is satisfied in a binary exponential family which is defined on n probability variables taking a binary value of either 1 or 0, and if their higher- order mutual interactions are taken into account. Due to this property, changes of correlation coefficients can be approximately predicted from the changes in the values of the parameters alone. In this paper, we study the way in which changes of the expectation parameters are related to changes in the elements of the Fisher information matrix and the normalized Fisher information matrix (negative partial correlation coefficients) in the expectation coordinate system that is dual to the natural parameter coordinate system in terms of the Fisher metric in the binary exponential family. Specifically, it can be shown that there exist monotonic properties of these statistical measures, the sufficient conditions for which are inclusion relations, in contrast to the similar relations between the correlation coefficients and natural parameters. By using this property, increases, decreases, and changes of the statistical measures can be predicted from the changes of the parameters without directly calculating the change of the measure itself. The properties of monotonic increase and decrease are also shown to be dual to each other. These properties are not direct conclusions derived from the properties in the natural coordinate system, but are newly obtained by analysis of the mixture family corresponding to the binary exponential family. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(12): 88–97, 1999