Φ - Divergences , Sufficiency , Bayes Sufficiency , and Deficiency

The paper studies the relations between φ-divergences and fundamental concepts of decision theory such as sufficiency, Bayes sufficiency, and LeCam’s deficiency. A new and considerably simplified approach is given to the spectral representation of φ-divergences already established in Österreicher and Feldman [28] under restrictive conditions and in Liese and Vajda [22], [23] in the general form. The simplification is achieved by a new integral representation of convex functions in terms of elementary convex functions which are strictly convex at one point only. Bayes sufficiency is characterized with the help of a binary model that consists of the joint distribution and the product of the marginal distributions of the observation and the parameter, respectively. LeCam’s deficiency is expressed in terms of φ-divergences where φ belongs to a class of convex functions whose curvature measures are finite and satisfy a normalization condition.