Nonextensive Statistical Mechanics

3.4 q-Generalization of the Kullback–Leibler Relative Entropy 85r q−1 − 1q − 1≥ 1 − 1 r= 1 − 1 r≤ 1 − 1 rif q > 0 ,if q = 0 , (3.156)if q < 0 .Consequently, for say q > 0, we have that[p(x)/p (0) (x)] q−1 − 1q − 1≥ 1 − p(0) (x)p(x), (3.157)hence∫dx p(x) [p(x)/p(0) (x)] q−1 − 1q − 1Therefore, we have∫≥[dx p(x) 1 − p(0) (x)p(x)]= 1 − 1 = 0 .(3.158)I q (p, p (0) ) ≥ 0 if q > 0 ,= 0 ifq = 0 , (3.159)≤ 0 ifq < 0 .It satisfies therefore the same basic property as the standard Kullback–Leibler entropy,and can be used for the same purposes, while we have now the extra freedomof choosing q adequately for the specific system which we are analyzing.By performing the transformation q − 1 2 ⇄ 1 − q into the definition (3.155), we2can easily prove the following property:I q (p, p (0) )q= I 1−q(p (0) , p)1 − q. (3.160)Consequently, as a family of entropy-based testing, it is enough to considerq ≥ 1/2, for which I q (p, p (0) ) ≥ 0 (the equality holding whenever p(x) = p (0) (x)almost everywhere). Also, as a corollary we have that only I 1/2 (p, p (0) ) is genericallysymmetric with regard to permutation between p and p (0) , i.e.,I 1/2 (p, p (0) ) = I 1/2 (p (0) , p) . (3.161)Moreover, the property I 1/2 (p, p (0) ) ≥ 0 implies∫dx √ p(x) p (0) (x) ≤ 1 . (3.162)