Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 47we can verify that, ∀q,S q =−D qW ∑i=1p xi | x=1 . (3.26)We consider this as an inspiring property, where the usual infinitesimal translationaloperation is replaced by a finite operation, namely, in this case, by the onewhich is basic for scale-invariance. Since the postulation of the entropy S q wasinspired by multifractal geometry, the least one can say is that this property is mostwelcome.3.2.2.7 Lesche-stability or Experimental RobustnessLet us start by emphasizing that this property is totally independent from concavity.For example, Renyi entropy SqR ≡ ln ∑ Wi=1 pq iis concave for 0 < q ≤ 1 and is1−qneither concave nor convex for q > 1. However, it is Lesche-unstable for all q > 0(excepting of course for q = 1) [79].It has been proved [110, 113] that the definition of experimental robustness, i.e.,Eq. (2.15), is satisfied for S q for q > 0 (See Fig. 3.8).3.2.2.8 Conditional Nonextensive Entropy, q-expectations Values, and EscortDistributionsLet us consider the entropy (3.18) and divide the set of W possibilities in Knonintersecting∑subsets, respectively, containing W 1 , W 2 ,...,W K elements, withKk=1 W k = W (1 ≤ K ≤ W ) [114]. We define the probabilitiesπ 1 ≡π 2 ≡π K ≡∑{W 1 terms}∑{W 2 terms}∑{W K terms}p i ,p i ,... (3.27)hence ∑ Kk=1 π k = 1. It is straightforward to verify the following property:S q ({p i }) = S q ({π k }) +p i ,K∑π q k S q({p i /π k }) , (3.28)where, consistently with Bayes’ formula, {p i /π k } are the conditional probabilities,and satisfy ∑ {W k terms} (p i/π k ) = 1(k = 1, 2,...,K ). Property (3.28) recovers, forq = 1, Shannon’s celebrated grouping relationk=1