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keithalewis
8 months ago
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Revisiting the Classics: Jensen's Inequality (2023...
A simpler definition of a convex function f is f(x) = sup { l(x) | l <= f where l is linear }.
If l <= f is linear then E[f(X)] >= E[l(X)] = l(E[X]). Taking the sup shows E[f(X)] >= f(E[X]).
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If l <= f is linear then E[f(X)] >= E[l(X)] = l(E[X]). Taking the sup shows E[f(X)] >= f(E[X]).