Characterizing Logarithmic Bregman Functions

TL;DR

This paper explores the mathematical properties of logarithmic Bregman functions, focusing on their relation to density power divergences and their potential for creating robust statistical inference methods.

Contribution

It characterizes the conditions under which logarithmic transformations of Bregman divergences relate to density power divergences, identifying limits for useful divergence families.

Findings

Logarithmic Bregman functions are closely related to density power divergences.

Such transformations are meaningful mainly within the Bregman divergence framework.

The study delineates the limits of divergence families derived from these transformations.

Abstract

Minimum divergence procedures based on the density power divergence and the logarithmic density power divergence have been extremely popular and successful in generating inference procedures which combine a high degree of model efficiency with strong outlier stability. Such procedures are always preferable in practical situations over procedures which achieve their robustness at a major cost of efficiency or are highly efficient but have poor robustness properties. The density power divergence (DPD) family of Basu et al.(1998) and the logarithmic density power divergence (LDPD) family of Jones et al.(2001) provide flexible classes of divergences where the adjustment between efficiency and robustness is controlled by a single, real, non-negative parameter. The usefulness of these two families of divergences in statistical inference makes it meaningful to search for other related families of divergences in the same spirit. The DPD family is a member of the class of Bregman divergences, and the LDPD family is obtained by log transformations of the different segments of the divergences within the DPD family. Both the DPD and LDPD families lead to the Kullback-Leibler divergence in the limiting case as the tuning parameter $\alpha \rightarrow 0$. In this paper we study this relation in detail, and demonstrate that such log transformations can only be meaningful in the context of the DPD (or the convex generating function of the DPD) within the general fold of Bregman divergences, giving us a limit to the extent to which the search for useful divergences could be successful.