Fisher's information and the class of f-divergences based on extended Arimoto's entropies

摘要

Csiszár (1963) and Ali and Silvey (1966) introduced the concept off-divergence, which is a measure of deviation of two probability distributions, like e.g. Pearson's χ_2-deviation, which is given in terms of a convex function f defined on [0,∞). Vajda (1972, 1973) extended Pearson's χ_2-deviation to the family of the so-called χ_α-divergences, α∈ [1, ∞). Österreicher and Vajda (2003) introduced a family of f-divergences closely linked to the class (Equation presented) of Arimoto's entropies (1971). Vajda (2009) extended the corresponding family Iφα off-divergences to all α ∈ ℝ. In Section 2 we consequently extend Arimoto's class of entropies to all α ∈ ℝ. Theorem 3 in Section 4 relates the family Iφα of f-divergences to Fisher's Information in a limiting way for all α ∈ ℝ except for those from a certain neighbourhood of α = 0. Its proof relies on an inequality of the form (Equation presented)which may be interesting also in its own right. This family of inequalities and its basic analytic counterpart are stated in Section 3. The corresponding proof is postponed to the Appendix. Section 2 was stimulated by Vajda's paper of 2009 and Section 4 considerably by his paper of 1973.