As far as I can tell, MCMC integration (e.g. VEGAS) is performed by sampling from a distribution proportional to $f(x)$ using MCMC, then building a density estimator $g(x)$ using these samples (for example, in case of VEGAS it is a factorized histogram $g(x_1, \dots, x_n) = g(x_1) \dots g(x_n)$), and then using this estimator to perform importance sampling and reweighting of values of $f(x)$, like $$I(f) = \mathbb E_{x \sim g(x)} \frac{f(x)}{g(x)}.$$ Correct me if I am wrong. It is not exactly clear to me whether one can use values $f_i$ at points $x_i \sim h(x) = \frac{f(x)}{\int f(x) dx}$ to directly approximate the final integral without introducing a density model $g(x)$ that unavoidably introduces an error into the estimate, especially in higher dimensions where density estimation is hard?

Let me give an example of an error I am talking about. Consider that $f(x)$ is just a standard 1D Gaussian pdf so it sums up to one. We receive three $(x_1, x_2, x_3)$ points near zero and corresponding values $(f_1, f_2, f_3)$ and then estimate $g_{K}(x)$ from these $x_i$ points using a kernel density estimator. Depending on the choice of the kernel width $K$ the density estimate in these points $g(x_i)$ would fluctuate significantly, i.e. for small $K$ values of $g(x_i)$ would be high and therefore $\sum_i \frac{f_i}{g(x_i)}$ would be low.

Here $K$ corresponds to our assumption about the smoothness of $f$ and if it is wrong, we get wrong results. The approximation used in VEGAS assumes that the function $f: R^n \to R$ factors well into the product of scalar functions.

Is there a way to avoid these possibly wrong assumptions? Assume that we already have plenty of values $(f_1, \dots , f_n)$ measured at points $(x_1, \dots , x_n)$ and we know that $x_i$ follow the distribution $x_i \sim h(x) \propto f(x)$?

  • $\begingroup$ The title is disconnected from your question. There is no MCMC element there. $\endgroup$ – Xi'an Feb 27 '19 at 7:15
  • $\begingroup$ @Xi'an mcmc is a way to get plenty of sample from a distribution proportional to $f(x)$, but it is not exactly clear to me how one can use these samples to approximate the final integral without introducing a density model $q_{\theta}(x)$ that minimizes the variance of the fraction $\frac{f(x)}{q(x)}$. I guess, the estimate of this fraction is the value of the integral itself. $\endgroup$ – MInner Feb 27 '19 at 17:26
  • $\begingroup$ This idea is interesting but kills the main property of importance sampling, which is to produce an unbiased estimator of the integral. $\endgroup$ – Xi'an Feb 28 '19 at 2:10
  • $\begingroup$ @Xi'an better now? $\endgroup$ – MInner Mar 1 '19 at 16:47
  • $\begingroup$ Your interpretation is not correct: MCMC simulations from $f$ are used directly in estimating the integral without an intermediate non-parametric density estimation, by virtue of the ergodic theorem. $\endgroup$ – Xi'an Mar 2 '19 at 2:39