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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 unifieddivergence 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