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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