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Is there a nice closed form expression for $\mathbb{E}_{\theta' \sim Dir(\alpha)} KL (Cat(x; \theta)|| Cat(x;\theta')$, where $Dir(\alpha)$ is the Dirichlet distribution with concentration parameters $\alpha$ and $Cat(x;\theta)$ is the discrete distribution with (log-)parameters $\theta$?

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It turns out I can answer this myself with some more effort. Write the inner KL expectation out explicitly, and it turns into the (negative) entropy of the categorical plus the cross-entropy between the categorical and the expectation of the log-Dirichlet. There's a nice closed form for the latter:

$\mathbb{E}_{X \sim Dir(\alpha)}[\log(X_i)] = \psi(\alpha_i) - \psi(\alpha_0)$ (from wikipedia).

Mafipulate a few terms and Bob's your uncle.

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