Expectation of Log Sum Dirichlet Random variables and Inference in Logistic Regression

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?

It has been argued that an analytical expression for the Expectation of Log sum of Dirichlet random variables is useful for inference in logistic regression. Can anyone explain why ? or give an explicit example (references) regression that would require the expectation Log sum of Dirichlet variables.

Answer 1

We have \begin{align} \log (\mathbf{c}^T \mathbf{X}) & = \log(\mathbf{c}^T)+ \log ( \mathbf{X})\\ \mathbb{E}[ \log (\mathbf{c}^T \mathbf{X})] & = E[\log(\mathbf{c}^T)+ \log ( \mathbf{X})]\\ & = E[\log(\mathbf{c}^T)]+ E[\log(\mathbf{X})]\\ & = \log(\mathbf{c}^T)+ E[\log(\mathbf{X})]\\ \end{align}