I want to compute an integral like this

$$\frac{\int_y g(y) e^{-\beta f(y)} \text{d} y } {\int_y e^{-\beta f(y)} \text{d} y}$$

where $f(y)$ is not necessarily convex and the dimension $d$ of $y$ is large.

This problem can be viewed as to integrate a function with respect to a density function whose normalized factor is unknown. It seems MCMC is a good choice? However for this general problem I didn't find any literature showing the convergence rate w.r.t the dimension $d$.

Remark: there are two ways I know to implement MCMC algorithms. One is by Metropolis–Hastings algorithm; another is by Metropolis-adjusted Langevin algorithm, which simulates a SDE $dX(t)=-\nabla f(x(t))dt + \sqrt{2\beta^{-1}} dB(t) $. In the latter case we also need to consider discretization errors of the SDE.

I guess without convexity assumptions on $f(y)$ the convergence rate could be $\mathcal{O}(e^{-d})$ slow. If this is true, it may suffer from the curse of dimensionality?

The reason why I came up with this problem is that we can just use simple Monte Carlo method, i.e. just sample uniformly distributed random variable to compute the integral on numerator and denominator separately. And as we know this time the variance of our estimate is not related to $d$.

I am very confused about the role MCMC plays in high dimension problems. Does someone there can help me figure out this? Any references are much appreciated.