I have a function $I_d(x)$ which defined over a plane. I could simulate the values of this function at different points. I have a ground truth probability density vector $p({\bf x})=(p_1(x),...,p_d(x))$ and my function is:
$I_d(x)=\sum_{i=1}^dp_i(x)\int_{A_i}f_i(x,a_i)da_i$
$\sum_{i=1}^{d}p_i(x)=1$
All I want to do is to estimate $p({\bf x})$. I define a likelihood function $q({\bf x}; \theta_1,...,\theta_d)=(q_1(x;\theta_i),...,q_d(x;\theta_d))$.
My question is how can I use gradient descent to reach the ground truth distribution by estimating $I_d(x)$ function based on $q({\bf x}; \theta_1,...,\theta_d)$?
