Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 3 / e / Solution

Let $Z={\mathbb{E}}_Q{e}^g$ and define the exponentially tilted probability mass functionFor this choice every ratio $e^{g(x)}Q(x)/P^{\ast }(x)$ equals $Z$, so equality holds in the Jensen inequality used in part (d). Henceand $P^{\ast }$ is the maximizer. This is the finite-alphabet Gibbs variational principle for relative entropy.

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