Next:Unified
Mutual InformationUp:Properties
of Unified Conditional
Previous:Biveriate
Cases Go to:Table
of Contents
Multivariate Cases
In this subsection, we shall extend the results studied in Section
6.2.1 for three or more random variables. These are given by the following
properties:
Property 6.8. We have
or
and![$\displaystyle \,\, 4).$](img1282.gif)
Property 6.9. We have
Property 6.10. We have
-
(i)
![$ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y,Z)\left\{\begin{array}{......Y),& r<0\ \mbox{with}\ s<1\ \mbox{or}\ 1<s<2-{1\over r}\end{array}\right.$](img1307.gif)
-
(ii)
for
=2
and 3;
The equality sign holds in (i) and (ii) iff
and
are independent given Z, i.e., iff
or
,
i,j,k.
Property 6.11. We have
Property 6.12. For all
,
we have
-
(i)
![$ {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y) \leq {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X) + {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y);$](img1312.gif)
-
(ii)
![$ {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y,Z) \leq {\ensuremath{\boldsymb......dsymbol{\mathscr{H}}}}_r^s(Y) + {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Z);$](img1313.gif)
-
(iii)
![$ \displaystyle {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_1,X_2,\cdots,x_\de......^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_i\vert X_1,X_2,\cdots,X_{i-1});$](img1314.gif)
-
(iv)
![$ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y\vert Z) \leq \, ^1{\ensuremath......}}}}_r^s(Y\vert X) +\, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X\vert Z).$](img1315.gif)
Property 6.13. If
then for all
,
we have
-
(i)
![$ d_r^s(X,Z) \leq d_r^s(X,Y) + d_r^s(Y,Z);$](img1317.gif)
-
(ii)
![$ \vert{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X) - {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y)\vert \leq {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y);$](img1318.gif)
-
(iii)
![$ \vert\,^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_1\vert Y_1) - \, ^1{\en......bol{\mathscr{H}}}}_r^s(X_1\vert Y_2)\vert \leq d_r^s(X_1,X_2) + d_r^s(Y_1,Y_2).$](img1319.gif)
Note 6.3. It is interesting to verify the properties 6.11, 6.12
and 6.13 for the measures
(
and
)
and
to find the conditions of their validites.
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil