integral theorems for Monte Carlo

Nhat Ho and Stephen G. Walker have just arXived a paper on the use of (Fourier) integral theorems for Monte Carlo estimators, following the earlier entry of Parzen: namely that for any integrable function,

$m(y)=\frac{1}{(2\pi)^d}\int_{\mathbb R^d}\int_{\mathbb R^d}\cos(s^\text{T}(y-x))m(x)\text dx\text ds$

which can be turned into an estimator of a density m based on a sample from m. This identity can be rewritten as

$m(y)=\lim_{R\to\infty}\frac{1}{\pi^d}\int_{\mathbb R^d}\prod_{i=1}^d\dfrac{\sin(R(y_i-x_i))}{y_i-x_i}\;m(x)\,\text dx$

and the paper generalises this identity to all cyclic functions. Even though it establishes that sin is the optimal choice. After reading this neat result, I however remain uncertain on how this could help with Monte Carlo integration.

This entry was posted on August 12, 2021 at 12:21 am and is filed under Books, pictures, Statistics with tags density estimator, Fourier transform, Joseph Fourier, Monte Carlo approximations, Riemann integration, Riemann sums, Texas, The University of Texas at Austin. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “integral theorems for Monte Carlo”

fhlb2357 Says:
August 12, 2021 at 12:09 pm

Ho and Walker shows how Fourier integrals lead to natural Monte Carlo estimators in an earlier paper:

Ho, N. and Walker, S.G., 2020. Multivariate smoothing via the Fourier integral theorem and Fourier kernel. arXiv preprint arXiv:2012.14482.

Reply

Xi'an's Og