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Unified (r,s)-Divergence Measures

In section 2.4 we studied the divergence measures known as Jensen difference divergence measure (or information radius) i.e. $I-$divergence and Jeffrey-Kullback-Leibler's $J-$divergence. Here our aim is to present different generalizations of these two measures having two scalar parameters. Their extension to $M-$class case is given in Chapter 5.

We see that the $I$ and $J-$divergence measures given respectively in (2.9) and (2.7) depend on the relative information, $D(P\vert\vert Q)$. Based on the unified expression ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$ and the expressions (2.9) and (2.7) we can generalize the $I$ and $J-$divergence measures. This we have done in the first generalization. An alternative approach to generalize the $I$ and $J-$divergence measures is also given, and it is based on the expressions appearing in the particular case of the first generalizations.