1
$\begingroup$

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 $x_i \sim h(x) \propto f(x)$?

|cite|improve this question
$\endgroup$

closed as unclear what you're asking by Xi'an, mkt, kjetil b halvorsen, mdewey, Michael Chernick Feb 28 at 2:33

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ The title is disconnected from your question. There is no MCMC element there. $\endgroup$ – Xi'an Feb 27 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 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 at 2:10
  • $\begingroup$ @Xi'an better now? $\endgroup$ – MInner Mar 1 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 at 2:39