Saturable Generalizations of Jensen's Inequality
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Jensen's inequality can be thought as answering the question of how knowledge of ๐ผ(X) allows us to bound ๐ผ(f(X)) in the cases where f is either convex or concave. Many generalizations have been proposed, which boil down to how additional knowledge allows us to sharpen the inequality. In this work, we investigate the question of how knowledge about expectations ๐ผ(f_i(X)) of a random vector X translate into bounds for ๐ผ(g(X)). Our results show that there is a connection between this problem and properties of convex hulls, allowing us to rewrite it as an optimization problem. The results of these optimization problems not only arrive at sharp bounds for ๐ผ(g(X)) but in some cases also yield discrete probability measures where equality holds. We develop both analytical and numerical approaches for finding these bounds and also study in more depth the case where the known information are the average and variance, for which some analytical results can be obtained.
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