Next:Mutual Information of DegreeUp:Entropy of Degree and
Previous:Entropy of Degree and Go to:Table of Contents

Conditional Entropies of Degree s

Based on the above definition of entropy of degree $ s$ the different ways of conditional entropies of degree $ s$ are as follows:

$\displaystyle ^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(X\vert Y)=\lef......\alpha}H^s_s(X\vert Y), & s \neq 1 \\  H(X\vert Y), & s=1\end{array}\right.$

for $ \alpha =1,2,3$ and $ 4$, where 
$\displaystyle ^1H^s_s(X\vert Y) =\,^2H^s_s(X\vert Y) = \sum_{j=1}^m p(x_j) H_s^s(X\vert Y=y_j),$
$\displaystyle ^3H^s_s(X\vert Y) = (2^{1-s}-1)^{-1} \left\{\left[\sum_{j=1}^m\......\sum_{i=1}^n p(x_i\vert y_j)^s\right)^{1/s}\right]^s -1\right\},\,\,s \neq 1 $
and 
$\displaystyle ^4H^s_s(X\vert Y) = H^s_s(X,Y) - H^s_s(Y)$
with
$\displaystyle H(X\vert Y) = -\sum_{j=1}^m p(y_j) \sum_{i=1}^n p(x_i\vert y_j)\,log\,p(x_i\vert y_j),$
and 
$\displaystyle H^s_s(X\vert Y=y_j) = (2^{1-s}-1)^{-1}\big[\sum_{i=1}^n p(x_i\vert y_j)^s -1\big], \,\, s \neq 1,$
for all$ j=1,2,\cdots,m.$

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