Problem
Given a following log-density function $$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$ where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters.
Find $\Bbb Cov( g(Y), h(Y))$ using $k(a,b)$.
My try
I believe that it got something to do with the score function and Fisher information. We basically can calculate the Fisher information using two methods:
- $\Bbb Var[(\ln p(y| a, b))'']$
- $\Bbb -E[(\ln p(y| a, b))']$
These are derivatives. My idea is to use both of these definitions and to make them equal. Since we have two parameters, I assume that it's a vector case. Fisher information using (1): $$\begin{bmatrix} \Bbb Var( g(Y))\\ \Bbb Var( h(Y)) \end{bmatrix}$$ And Fisher infomration using (2) $$\begin{bmatrix} k_a''\\ k_b'' \end{bmatrix}$$ Even though I've found each of the variances, I have them separately and don't know how to get to the covariance.
