I'm currently trying to evaluate an integral with respect to an MCMC kernel, and I wanted someone else to read what I `proved' to see if it's really correct. It just doesn't feel right.
Suppose I have an MCMC Kernel K(a'|a) with stationary distribution $p(a)$. I want to take the expectation of a function $f(a)$ given a point a' through the kernel like so,
\int_{a} f(a) K(a|a')
I use detailed balance,
= \int_{a} f(a) \frac{K(a'|a) p(a)}{p(a')} = \mathbb{E}[f(a)K(a'|a)] \frac{1}{p(a')}
Independence (at least I think they should be independent?) = \mathbb{E}[f(a)] \mathbb{E}[K(a'|a)] \frac{1}{p(a')}
And detailed balance once again
= \mathbb{E}[f(a)] p(a') \frac{1}{p(a')}
Now this just seems incorrect. If I interpret this correctly, it's telling me that if I take a large number of samples from K(a|a') from any starting point a' and integrate a function wrt to it, I might as well have had the original distribution. I must have done something wrong.