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Introduction

The concept of the generalized entropy measures discussed discussed in Chapter 1 needs to be developed for multivariate probability distributions - in particular for bivariate cases, especially for problems of communication that require the analysis of messages sent over a channel and received at the other end. The same is also required in bounding bayesian probability of error, comparision of experiments, etc.. Following the notations of Chapter 1, section 1.4, the joint and individual unified (r,s)-entropies can be written as ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y)$, ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X)$ and ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X\vert Y)$, where ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s$ is the unified$(r,s)-$entropy given by (3.8)

There is no unique way to define the conditional generalized entropy. It has been defined in different ways by different authors. We shall specify here four different ways to define the conditional generalized entropy. We shall observe that these approaches in the limiting case reduce to the well-known Shannon's conditional entropy given in Chapter 1, section 1.4. The idea of mutual information has also been generalized for the unified $(r,s)-$entropy measures.

Henceforth, unless otherwise specified, the letters$X$, $Y$, $Z$, ..., etc., will represent discrete, finite random variables.