- #1
bayesian
- 2
- 0
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].
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?
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?