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## Unified (r,s)-Mutual Information

In this section, we present six different ways to define the unified mutual information.

Let us consider and (6.6)
and and (6.7)
where (6.8)
and etc. We call the measures ( and ) the unified mutual information among the random variables and . The measures ( and ) we call the unified mutual information among the random variables and given .

Property 6.14. We have

(i) (resp. ) ( and ), with equality iff and are independent.
(ii) (resp. ) ( and ), with equality iff and are independent given .
The above property 6.12 holds under the following conditions:
( ) For  with or (resp. with or );
( ) For and 3, (resp.  );
( ) For  (resp. ) (only for part(i)).
Note 6.4. Similar to Shannon's entropy (property 1.51) we left for the readers to check the validity of the following inequality: i.e., to find the conditions on the parameters and for the validity of the above expression.

Property 6.15. We have and (6.9)
Note 6.5. The relation (6.9) is famous as "additive property". It also hold when , but at moment, we are not aware of the conditons on and for which is nonnegative, but the particular case when is discussed in section 6.4

For Simplicity, let us write and forall .

We have  where is as given in (2.1)

We can write where for all Thus The two more ways to define the unified mutual information are given by (6.10) (6.11)
where is as given by (4.1).

Property 6.16. For and , we have

(i) (ii) with equality iff and are independent.
(iii) Note 6.6. If we consider the fifth and sixth way of unified conditional entropies in the following way (6.12)
then, we don't know for what values of and the measures given in (6.12) turn out to be nonnegative.

Note 6.7. Some of the above generalized mutual information measures can be connected to unified divergence measures given in Chapter 5. These connections are as follows:

In (5.12), take and , we get Similarly, we can write  In the notes 6.4 and 6.5, we have seen that there are difficulties in getting certain values of and for which the measures or and ) become nonnegative. But, in the particular case, when , the above difficulties are overcomed. The particular case for is discussed in the following subsection.

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