One form of Jensen's inequality is
If $X$ is a random variable and $g$ is a convex function, then $\mathbb{E}(g(X))\geq g(\mathbb{E}(X))$.
Just out of curiosity, when do we have equality? If and only if $g$ is constant?
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RobPratt
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11Also if $g$ is affine. – hardmath Jan 05 '14 at 19:40
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10And/or $X$ constant almost surely. – Did Jan 05 '14 at 19:45
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5I think an if and only if condition would be: there is equality iff there exists an interval A such that g is affine on A, and X belongs to A almost surely. – D. Thomine Nov 20 '14 at 22:01
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2https://math.stackexchange.com/questions/1160095/convexity-and-equality-in-jensen-inequality – StubbornAtom May 15 '20 at 14:39
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An example of X being constant almost surely is KL(p||q)=0 iff p=q almost surely. Here g is -ln() so can't be affine. The constant must be one (almost surely) due to normalization. – user5280911 Mar 15 '22 at 07:18
1 Answers
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Just for the sake of having an "answered" question (thanks to @hardmath and @Did), Jensen's inequality is equality when $g$ is affine or $X$ is constant almost surely.
Jacob A
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