One KL-difference identity plus non-negativity of KL derives convexity of the log-partition function, Gibbs variational principle, Pythagorean theorems, and tilting formulas for exponential families.
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Exponential families from a single KL identity
One KL-difference identity plus non-negativity of KL derives convexity of the log-partition function, Gibbs variational principle, Pythagorean theorems, and tilting formulas for exponential families.