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Divergence Function, Duality, and Convex Analysis

junz{at}umich.edu, Department of Psychology, University of Michigan, Ann Arbor, MI 48109, U. S. A

From a smooth, strictly convex function : Rⁿ R, a parametric family of divergence function ⁽⁾ may be introduced:

for x, y int dom () Rⁿ, and for R, with ^{(± 1)} defined through taking the limit of . Each member is shown to induce an -independent Riemannian metric, as well as a pair of dual -connections, which are generally nonflat, except for = ± 1. In the latter case, ^{(± 1)} reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate ^* and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; Amari & Nagaoka, 2000). This formulation based on convex analysis naturally extends the information-geometric interpretation of divergence functions (Eguchi, 1983) to allow the distinction between two different kinds of duality: referential duality ( - ) and representational duality ( ^*). When applied to (not necessarily normalized) probability densities, the concept of conjugated representations of densities is introduced, so that ±-connections defined on probability densities embody both referential and representational duality and are hence themselves bidual. When restricted to a finite-dimensional affine submanifold, the natural parameters of a certain representation of densities and the expectation parameters under its conjugate representation form biorthogonal coordinates. The alpha representation (indexed by ß now, ß [-1,1]) is shown to be the only measure-invariant representation. The resulting two-parameter family of divergence functionals ^{(, ß)}, (, ß) [-1,1] x [-1,1] induces identical Fisher information but bidual alpha-connection pairs; it reduces in form to Amari's alpha-divergence family when = ± 1 or when ß=1, but to the family of Jensen difference (Rao, 1987) when ß= -1.

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