# Monte-Carlo Estimation of conditional expectation term

I want to ask if my approach to estimation of the following quantity is correct:

I have $$n$$ i.i.d. draws $$\{(X_i,Z_i) \}_{i=1}^n$$ and I want to estimate for a fixed $$(i,j)$$ pair the quantity: $$\mathbb{E}[f(X_i,X_j,Z_i,Z_j)|Z_i,Z_j]$$

where $$f$$ is some function and I have access to the distribution $$P_{X|Z}$$. My approach is to generate $$M$$ extra samples of $$\{\tilde{X}^{(m)}_i\}_{m=1}^M$$ from $$P_{X|Z=Z_i}$$ and similarly M extra samples of $$\{\tilde{X}^{(m)}_j\}_{m=1}^M$$ from $$P_{X|Z=Z_j}$$. I then use these extra samples to build the following estimator:

$$\mathbb{E}[f(X,X',Z,Z')|Z,Z] \approx \frac{1}{M}\sum_{m=1}^M f(\tilde{X}_j^{(m)}, \tilde{X}_j^{(m)}, Z_i, Z_j).$$

This seems like the intuitive thing to do, but I am wondering if there any issues with it?

• This seems correct! – Xi'an Jan 15 at 16:45