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Properties of Information Measures

Some properties of the measures (2.1) and (2.2) are presented in this subsection. For simplicity, let us denote and .

Property 2.1. (Continuity).  and are continuous functions of the pair .

Property 2.2. (Symmetry). and are symmetric functions of their arguments in pair , i.e., for and 2, we have where is any permutation from to .

Property 2.3. (Expansibility). For and 2, we have Property 2.4. (Additivity). For and 2, we have where Property 2.5. (Sum representation). We can write and where and for all Property 2.6. (Nonnegativity). and , with equality iff for and  for .

Property 2.7. (Recursivity). For and , we have   and .

Property 2.8. (Strongly additive). For =   ,  , we have  Property 2.9. (Functional equation). Let then for and 2, we have
(i)  for all with .
(ii) where Property 2.10. (Parallelogram identity). For any , ,  , we have Property 2.11. (Convexity). We have
(i) is a convex function in a pair of distributions  .
(ii) is a convex function in .
Property 2.12. (Data processing inequality). Let and be two probability distributions, where   ; is a stochastic matrix such that  . Then Property 2.13. (Schur-convexity). We have
(i) is a Schur-convex function in the pair .
(ii) is a Schur-convex function in .
Property 2.14. (Inequality). We have with equality iff .

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil