Nonextensive Pythagoras' Theorem

Abstract

Kullback-Leibler relative-entropy, in cases involving distributions resulting from relative-entropy minimization, has a celebrated property reminiscent of squared Euclidean distance: it satisfies an analogue of the Pythagoras' theorem. And hence, this property is referred to as Pythagoras' theorem of relative-entropy minimization or triangle equality and plays a fundamental role in geometrical approaches of statistical estimation theory like information geometry. Equvalent of Pythagoras' theorem in the generalized nonextensive formalism is established in (Dukkipati at el., Physica A, 361 (2006) 124-138). In this paper we give a detailed account of it.