f -DIVERGENCES - REPRESENTATION THEOREM AND METRIZABILITY

In this talk we are first going to state the so-called ’Representation Theorem’ which provides the representation of general f -divergences in terms of the class of elementary divergences. Then we consider the risk set of a (simple versus simple) testing problem and its characterization by the above mentioned class. These ingredients enable us to prove a sharpening of the ’Range of Values Theorem’ presented in Talk 1. In the sequel, necessary and sufficient conditions are given so that f -divergences allow for a topology, respectively a metric. In addition, sufficient conditions are given which permits a suitable power of an f divergence to be a distance. Finally, we investigate the classes of f -divergences discussed in Talk 1 along these lines. This talk was presented while participating in a workshop of the Research Group in Mathematical Inequalities and Applications at the Victoria University, Melbourne, Australia, in November 2002. Let Ω = {x1, x2, ...} be a set, P(Ω) the set of all subsets of Ω , P the set of all probability distributions P = (p(x) : x ∈ Ω) on Ω and P2 the set of all (simple) testing problems. Furthermore, let F0 be the set of convex functions f : [0,∞) 7→ (−∞,∞] continuous at 0 and satisfying f(1) = 0 , f∗ ∈ F0 the ∗-conjugate (convex) function of f and f̃ = f + f∗ . 1 REPRESENTATION THEOREM 1.1 REPRESENTATION BY ELEMENTARY DIVERGENCES At the beginning we investigate the f -divergences of the Class (IV) of Elementary Divergences given by ft(u) = max(u− t, 0) , t ≥ 0