MCMC: Integral Approximation?

Question

Let's say I want to approximate the following integral: $$I = \int_0^5 R(x)f(x)dx$$ where $R(x)$ is a Rayleigh distribution and $f(x)$ is some generic function.

I generate $n$ samples using Metropolis Hastings MCMC.

Now is the approximation simply: $$I' = \frac{1}{n}\sum_{i=1}^nf(x_i)$$ where the individual samples $x_i$ are generated from Metropolis Hastings?

Is the target pdf in Metropolis Hastings MCMC only $R(x)$? or the entire function?

Any generic background/resources concerning the defining features of MCMC would be great!

Thanks!

Answer 1

The generating PDF for $x_i$s in the MH algorithm must be the Truncated Rayleigh distribution since your integration bounds are limited. So you must actually have $$f_{x_i}(x)={R(x)\over \Pr\{X<5\}}$$where $f_{x_i}(x)$ is the PDF of $x_i$ and $X$ is a Rayleigh (not truncated) random variable.