Get covariance from log-density function

Question

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:

1. $\Bbb Var[(\ln p(y| a, b))'']$
2. $\Bbb -E[(\ln p(y| a, b))']$

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.

Answer 1

Here is how I think $\Bbb{Cov}(g(Y),h(Y))$ is calculated (and $\Bbb{Var}(g(Y)), \Bbb{Var}(h(Y))$)

$\Bbb{Var}(g(Y))\\ =\Bbb{Var}(g(Y))+2\cdot0+0\\ =\Bbb{Var}(g(Y))+2\cdot\Bbb{Cov}(g(Y),k'_a)+\Bbb{Var}(k'_a)\\ =\Bbb{Var}(g(Y)+k'_a)\\ =\Bbb{Var}(\partial_a \ln(p(x|a,b))\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)-\Bbb{E}(\partial_a \ln(p(x|a,b))]^2)\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)]^2)-[\Bbb{E}(\partial_a \ln(p(x|a,b)))]^2\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)]^2)-0^2\\ =-\Bbb{E}(\partial^2_a \ln(p(x|a,b))\\ =-\Bbb{E}(k''_a)\\ =-k''_a\\ $

$\Bbb{Var}(h(Y))\\ =\Bbb{Var}(h(Y))+2\cdot0+0\\ =\Bbb{Var}(h(Y))+2\cdot\Bbb{Cov}(h(Y),k'_b)+\Bbb{Var}(k'_b)\\ =\Bbb{Var}(h(Y)+k'_b)\\ =\Bbb{Var}(\partial_b \ln(p(x|a,b))\\ =\Bbb{E}([\partial_b \ln(p(x|a,b)-\Bbb{E}(\partial_b \ln(p(x|a,b))]^2)\\ =\Bbb{E}([\partial_b \ln(p(x|a,b)]^2)-[\Bbb{E}(\partial_b \ln(p(x|a,b)))]^2\\ =\Bbb{E}([\partial_b \ln(p(x|a,b)]^2)-0^2\\ =-\Bbb{E}(\partial^2_b \ln(p(x|a,b))\\ =-\Bbb{E}(k''_b)\\ =-k''_b\\ $

$\Bbb{Cov}(g(Y),h(Y))\\ =\Bbb{Cov}(g(Y),h(Y))+0+0+0\\ =\Bbb{Cov}(g(Y),h(Y))+\Bbb{Cov}(k'_a,h(Y))+\Bbb{Cov}(g(Y),k'_b)+\Bbb{Cov}(k'_a,k'_b)\\ =\Bbb{Cov}(g(Y)+k'_a,h(Y)+k'_b)\\ =\Bbb{Cov}(\partial_a \ln(p(x|a,b),\partial_b \ln(p(x|a,b))\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)-\Bbb{E}(\partial_a \ln(p(x|a,b))][\partial_b \ln(p(x|a,b)-\Bbb{E}(\partial_b \ln(p(x|a,b))])\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)][\partial_b \ln(p(x|a,b)])-\Bbb{E}([\partial_a \ln(p(x|a,b)])\Bbb{E}([\partial_b \ln(p(x|a,b)])\\ =\Bbb{E}([\partial_a \ln(p(x|a,b)][\partial_b \ln(p(x|a,b)])-0\cdot0\\ =-\Bbb{E}(\partial_a\partial_b \ln(p(x|a,b)])\\ =-\Bbb{E}(k''_{ab})\\ =-k''_{ab} $