I am struggling to understand what does the following expression evaluate to.
"I am looking for general answer, not actual evaluation - i.e is dirac dealta making the integral center around h(x)? what exatly is happeneing when dirac delta is being introduced"
$$\int g(\mathbf{x})p(x_1,y_1)\delta(x_1 - h(\mathbf x))dx_1dy_1 $$
It has been taken from the following context: $${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP
$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$