Nonextensive Statistical Mechanics

86 3 Generalizing What We LearntThis expression can be interpreted as the continuous version of the scalar productbetween two unitary vectors, namely √ p(x) and √ p (0) (x), and is directly related tothe so-called Fisher genetic distance [89].Let us also q-generalize Eq. (2.37). By choosing as p (0) (x) the uniform distributionon a compact support of length W , we easily establish the desired generalization,15 i.e.,I q (p, 1/W ) = W q−1 [ln q W − S q (p)] . (3.163)As in the q = 1 case, for q > 0, the minimization of the q- generalized Kulback–Leibler entropy I q may be used instead of the maximization of the entropy S q .Moreproperties can be found in [92].Let us finally mention an elegant property, referred to as the triangle pseudoequality[95, 96]. Through some algebra, it is possible to proveI q (p, p ′ ) = I q (p, p ′′ ) + I q (p ′′ , p ′ ) + (q − 1)I q (p, p ′′ )I q (p ′′ , p ′ ) . (3.164)A simple corollary follows, namelyI q (p, p ′ ) ≥ I q (p, p ′′ ) + I q (p ′′ , p ′ ) ifq > 1 ,= I q (p, p ′′ ) + I q (p ′′ , p ′ ) ifq = 1 , (3.165)≤ I q (p, p ′′ ) + I q (p ′′ , p ′ ) ifq < 1 .The name triangle pseudo-equality for Eq. (3.164) obviously comes from theq = 1 case, where we do have a strict equality.Let us now adapt our present main result, i.e., Eq. (3.159), to the problem of independenceof random variables. Let us consider the two-dimensional random variable(x, y), and its corresponding distribution function p(x, y), with ∫ dx dy p(x, y) =1. The marginal distribution functions are then given by h 1 (x) ≡ ∫ dy p(x, y) andh 2 (y) ≡ ∫ dx p(x, y). The discrimination criterion for independence concerns thecomparison of p(x, y) with p (0) (x, y) ≡ h 1 (x) h 2 (y). The one-dimensional randomvariables x and y are independent if and only if p(x, y) = p (0) (x, y) (almost everywhere).The criterion (3.159) then becomes∫dx dy p(x, y)[p(x,y)h 1 (x) h 2 (y)] q−1− 1q − 1≥ 0 (q ≥ 1/2) . (3.166)In the limit q → 1, this criterion recovers the usual one, namely [90]15 This formula appears misprinted in Eq. (3.15) of the original paper [88]. This erratum was kindlycommunicated to me by R. Piasecki.