I need to compute the Fisher information matrix of the Dirichlet distribution, defined in the following way:
$$r(P; \pi, \rho) = \frac{\Gamma(c)}{\prod_{i=1}^{k} \Gamma(c \pi_i)} \cdot \prod_{i=1}^{k} P_i^{c\pi_i - 1},$$
where $c=\rho^{-2}(1-\rho^2) = \sum_{i=1}^{k} \alpha_i$. I defined the log likelihood as: $$\log\Gamma(c) - \sum_{i=1}^{k} \log\Gamma(c\pi_i) + \sum_{i=1}^{k}(c\pi_i-1) \log P_i,$$ and I hope it is correct. From here I have that the first derivative of the log likelihood with respect to $\pi_i$ is $-c\psi(c\pi_i)+clogP_i$ and the second derivative is $-c^2 * \psi'(c\pi_i)$.
I have to compute the $E_P[-d^2l/d\pi_i^2]$ and I am given the result of this which is equal to $c^2[\psi'(c\pi_i) + \psi'(c\pi_k)]$ for $i = 1,...,k-1$. But how can I get this result? I don't get why this is the result... from where they took $\psi'(c\pi_k)$?
And also which is the expectation of a digamma and trigamma function?
Online it is hard to find this kind of information.
Thanks a lot again if you can help me!