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Bivariate Cases
In this subsection, we shall give properties of the generalized
conditional entropies for the bivariate case.
Property 6.1. For
,
we have
or![$\displaystyle \ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y).$](img1280.gif)
Property 6.2. We have
and
Property 6.3. If
and
are independent random variables, then
![$\displaystyle {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y)={\ensuremath{\bo......ldsymbol{\mathscr{H}}}}^s_r(X)\,{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y).$](img1283.gif)
Property 6.4. For any
and
,
we have
Property 6.5. The following inequalities hold:
-
(i)
![$ ^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y),& r<0\end{array}\right.$](img1286.gif)
-
(ii)
![$ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......remath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y),& s\leq r\end{array}\right.$](img1287.gif)
-
(iii)
![$ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y)\left\{\begin{array}{ll} \leq ......ert Y),& r{{s-1}\over {r-1}}\leq 1\leq {{s-1}\over {r-1}}\end{array}\right.$](img1288.gif)
-
(iv)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......rt Y),& r{{s-1}\over {r-1}}\leq 1\leq{{s-1}\over {r-1}}\end{array}\right.$](img1289.gif)
Note 6.2. The conditions
are equivalent to either
or
,
,
and the conditions
are equivalent to either
,
or
.
Property 6.6. We have
-
(i)
![$ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......\ \mbox{with}\s\leq 1\ \mbox{or}\ 1<s\leq 2-{1\over r};\end{array}\right.$](img1296.gif)
-
(ii)
for
=2
and 3;
-
(iii)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\left\{\begin{array}{ll......suremath{\boldsymbol{\mathscr{H}}}}^s_r(X),& s\leq r < 0.\end{array}\right.$](img1298.gif)
The equality sign holds iff
and
are independent random variables.
As a consequence of properties 6.5 and 6.6, we have the following property:
Property 6.7. (i) For
,
we have
![$\displaystyle 0\leq {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X)\leq\ ^3{\ens......}}}^s_r(X\vert Y)\leq\ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y).$](img1299.gif)
(ii) For
,
we have
![$\displaystyle 0 \leq \, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)......athscr{H}}}}^s_r(X\vert Y)\leq{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X).$](img1300.gif)
(iii) For
,
we have
![$\displaystyle 0\leq \ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\l......athscr{H}}}}^s_r(X\vert Y)\leq{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X).$](img1301.gif)
(iv) For
,
we have
![$\displaystyle 0\leq \ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y)\l......}}^s_r(X\vert Y)\leq\^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y).$](img1303.gif)
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil