I am trying to estimate an unknown distribution say, $p(x)$, using MH-Algorithm.
Assume that there is a function $f(x) \propto p(x)$,
However, the exact $f(x)$ is also unknown, I can only compute its unbiased estimator say $\hat{f}(x)$, i.e., $E(\hat{f}(x))=E(f(x))$ for all $x$.
Is there any related theoretical results that can analyze the convergence rate or stability for such method?