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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:

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

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