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Properties of Conditional Entropies
of Degree s
We have the following properties:
Property 6.17. For all
,
we have
-
(i)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s((X,Y)\wedge Z) = \, ^4{\ensurema......wedge Y) + \, ^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s(Y \wedge Z\vert X);$](img1388.gif)
-
(ii)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s(X \wedge Y) = {\ensuremath{\bold......r{H}}}}_s^s(X,Y) = \,^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(Y \wedge X);$](img1389.gif)
-
(iii)
![$ {\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(X) + {\ensuremath{\boldsymbol{\mat......((X,Y) \wedge Z) + \,^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s(X \wedge Y);$](img1390.gif)
-
(iv)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s((X,Y) \wedge Z\vert V) = \,^4{\e......ert V) + \, ^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s(Y \wedge Z\vert X,V).$](img1391.gif)
Property 6.18. For all
,
we have
with equality iff
and
are independent given
.
Property 6.19. For all
,
we have
-
(i)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(X\vert Z) \leq \, ^4{\ensuremath......}}}_s^s(X\vert Y) + \, ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(Y\vert Z);$](img1393.gif)
-
(ii)
![$ ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(X,Y\vert Z) \leq \, ^4{\ensurema......}}}_s^s(X\vert Z) + \, ^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(Y\vert Z);$](img1394.gif)
-
(iii) If
,
then
Property 6.20. Let
and
then, we have
-
(i)
and![$ \,\, 3;$](img1401.gif)
-
(ii)
![$ \vert{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(X) - {\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(Y)\vert \leq d_s^1(X,Y);$](img1402.gif)
-
(iii)
![$ \vert\,^4{\ensuremath{\boldsymbol{\mathscr{H}}}}_s^s(X_1\vert Y_1) -\, ^4{\ens......H}}}}_s^s(X_2\vert Y_2)\vert \leq d_s^1(X_1,X_2) + d_s^1(Y_1,Y_2), \, s \geq 1;$](img1403.gif)
-
(iv)
![$ \vert\,^4{\ensuremath{\boldsymbol{\mathscr{F}}}}_s^s(X_1 \wedge Y_1) -\, ^4{\e......}}}_s^s(X_2 \wedge Y_2)\vert \leq d_s^1(X_1,X_2) + d_s^1(Y_1,Y_2), \, s \geq 1.$](img1404.gif)
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil