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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?

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