# Expectation of KL between Categorical and samples from a Dirichlet

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 $$\theta$$?

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