# Variance estimates for a huge number of estimates

I'm estimating a finite number ($$\approx1e5$$) of integrals $$\lambda g$$ using the Metropolis-Hastings algorithm with target distribution $$\mu=\frac{p\lambda}c$$ (where $$c:=\lambda p\in(0,\infty)$$) and the importance sampling estimator.

In each iteration, given the current state $$x$$, I need to compute the quantity $$\int\lambda({\rm d}y)\frac{\left|1_H(y)p(y)-\frac{p(y)}c\lambda(1_Hp)\right|^2}{r(x,y)}\tag1,$$ where $$r(x,\;\cdot\;)$$ is a probability density with respect to $$\lambda$$ and $$H\in\mathcal E$$ with $$\lambda(H)\in(0,\infty)$$.

I'm able to sample from $$r(x,\;\cdot\;)\lambda$$ and hence can estimate $$(1)$$ by ordinary Monte Carlo integration. However, since I need to compute $$(1)$$ for $$\approx1e5$$ different $$H$$ in each iteration of the Metropolis-Hastings algorithm, this approach is extremely slow (practically unusable).

Is there a better approach for a problem of this kind? If not: One of the biggest problems with the Monte Carlo approach is that I have no clue which sample size I should choose for my estimate of $$(1)$$. Is there a good rule which tells me that enough samples have been taken?