Suppose I can estimate a distribution $G(x)$ as

$$ \hat G(x) = f(x, X)$$

where $X$ are my data points and $f$ is a known, but computationally heavy function.

Eventually I'm interested in

$$ h(\theta) = \int k(x, \theta) dG(x)$$

which I will have to evaluate many many times for different $\theta$. Thus, sampling from $G(x)$ repeatedly is very slow, since it involves repeated evaluation of $f(x,X)$. 

I'm thinking about pre-evaluating $\hat G(x)$ at a set of pre-determined points, e.g., some grid, and only sampling from there, but I'm worried that this might bias my results.

What can be done here?