Nonextensive Statistical Mechanics

3.4 q-Generalization of the Kullback–Leibler Relative Entropy 87∫∫∫dx dy p(x, y) lnp(x, y) − dx h 1 (x)lnh 1 (x) − dyh 2 (y)lnh 2 (y) ≥ 0 .(3.167)For q = 1/2, we obtain a particularly simple criterion, namely∫dx dy p(x, y) √ p(x, y) h 1 (x) h 2 (y) ≤ 1 . (3.168)For q = 2, we obtain∫dx dy[p(x, y)]2h 1 (x) h 2 (y) ≥ 1 . (3.169)This can be considered as a satisfactory quadratic-like criterion, as opposed to thequantity introduced in [91]. We refer to the quantity frequently used in economics[91], namely, for h 1 = h 2 ≡ h,∫dx dy[p(x, y)] 2 −{ ∫ dx [h(x)] 2} 2. (3.170)This quantity has not a definite sign. In fact, if x and y are independent, thisquantity vanishes. But, if it vanishes, x and y are not necessarily independent. Inother words, its zero is not a necessary and sufficient condition for independence,and therefore it does not constitute an optimal criterion. It could be advantageouslyreplaced, in applications such as financial analysis, by the present criterion (3.169).The generalization of criterion (3.166) for an arbitrary number d of variables(with d ≥ 2) is straightforward, namelyI q (p(x 1 , x 2 ,...,x d ), p (0) (x 1 , x 2 ,...,x d )) ≥ 0 (q ≥ 1/2) , (3.171)where[ ∫p (0) (x 1 , x 2 ,...,x d ) ≡ dx 2 dx 3 ...dx d p(x 1 , x 2 ,...x d )][ ∫× dx 1 dx 3 ...dx d p(x 1 , x 2 ,...x d )]× ...[ ∫× dx 1 dx 2 ...dx d−1 p(x 1 , x 2 ,...x d )]. (3.172)Depending on the specific purpose, one might even prefer to use the symmetrizedversion of the criterion, i.e.,