Expectation of log of linear function of the Dirichlet distribution

Question

Given $\mathbf{X}\sim\mathsf{Dir}(\alpha_1,\cdots,\alpha_k)$, is there an expression for the expectation $$ \mathbb{E}\left[ \log \left(\mathbf{c}^\top \mathbf{X} \right)\right] $$ where $\mathbf{c}\in\mathbb{R}_+^k$ is a vector of positive constants?

Answer 1

I think there isn't a good way to express this; for what I've calculated, it comes down to the fact that the sum of independent Gamma random variables isn't a nice distribution when the shape and scale are both different. Here's what I've got:

If we set $Y_i = \text{Gamma}(\alpha_i,1)$, then we know that $$\mathbf{X} \sim \left(\frac{Y_1}{\sum Y_i}, \ldots, \frac{Y_k}{\sum Y_i} \right).$$

This lets us calculate: \begin{align*} \mathbb{E}[\log(\mathbf{c}^T \mathbf{X})] &= \mathbb{E}\left[\log\left(\sum c_i Y_i\right)\right] - \mathbb{E}\left[\log \left(\sum Y_i\right) \right]. \end{align*}

We know that $\sum Y_i \sim \text{Gamma}(\sum \alpha_i,1)$, implying that $$\mathbb{E}\left[\log \left(\sum Y_i\right) \right] = \psi(\sum \alpha_i)$$ where $\psi$ is the digamma function. I don't think there's a nice way to deal with the first term because we can't say a lot about the distribution of $\sum c_i Y_i$, so I think this is as far as you can get.