This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model.
We all know, given that $\boldsymbol{\theta} \sim \mathrm{Dir}(\boldsymbol{\alpha})$, then $E_{p(\boldsymbol{\theta} \mid \boldsymbol{\alpha})}[\log{\theta_k}] = \Psi(\alpha_k) - \Psi(\sum_{k'=1}^K \alpha_{k'})$, where $\Psi$ is digamma function.
Then, how to calculate the following expectation: $$E_{q(\psi, \boldsymbol{\varphi} \mid \boldsymbol{\lambda}, \boldsymbol{\mu})}[\log{((1-\psi)\cdot \varphi_{v} +\psi)}]$$ where $\psi$ and $\boldsymbol{\varphi}$ are independent, $\psi \sim \mathrm{Beta}(\lambda_1, \lambda_2), \boldsymbol{\varphi} = (\varphi_1, \varphi_2,\cdots, \varphi_V) \sim \mathrm{Dir}(\mu_1, \mu_2, \dots,\mu_V)$
