Given two (dependent) random variables [itex]X[/itex] and [itex]Y[/itex] with joint PDF [itex]p(x,y)[/itex] [itex]=p(x|y)p(y)[/itex] [itex]=p(y|x)p(x)[/itex], let [itex]H[X][/itex] be real-valued concave function of [itex]p(x)[/itex], and [itex]H[X|Y][/itex] the expectation of [itex]H[/itex] of [itex]p(x|y)[/itex] with respect to [itex]p(y)[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Examples of possible functions [itex]H[/itex] include the entropy of [itex]X[/itex], or its variance.

The concavity of [itex]H[/itex] implies that [itex]H[X]-H[X|Y]≥0[/itex] (through Jensen's inequality).

Question:

What additional conditions (if any) on [itex]H[/itex] are imposed if we in addtion require that [itex]H[X]-H[X|Y][/itex] should also be concave with respect to [itex]p(x)[/itex], if [itex]p(y|x)[/itex] remains fixed?

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# On concave functions over spaces of probabilty distributions

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