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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12$\begingroup$ Also if $g$ is affine. $\endgroup$– hardmathJan 5, 2014 at 19:40
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10$\begingroup$ And/or $X$ constant almost surely. $\endgroup$– DidJan 5, 2014 at 19:45
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6$\begingroup$ I 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. $\endgroup$– D. ThomineNov 20, 2014 at 22:01
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2$\begingroup$ math.stackexchange.com/questions/1160095/… $\endgroup$– StubbornAtomMay 15, 2020 at 14:39
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$\begingroup$ 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. $\endgroup$– user5280911Mar 15, 2022 at 7:18
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1 Answer
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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.
answered Nov 20, 2014 at 21:59