On numerical calculus of probability density functions

Gregory Beylkin
University of Colorado Boulder

While the probability density function (PDF) of the sum of two independent random variables is easily described as the convolution of their PDFs, the expression for the PDF of their product is significantly more complicated. As far as we know, the only universal method currently available for this purpose relies on a Monte-Carlo (MC) approach, where one samples the individual PDFs, computes the products, and collects enough samples to achieve a certain accuracy. However, due to the slow convergence of MC type methods, achieving high accuracy is not feasible.

By introducing a new approximate multiresolution analysis (MRA) which uses a single Gaussian as the scaling function (we call it Gaussian MRA or GMRA), we can now rapidly compute the PDF of the product of two independent random variables with a full accuracy control. In contrast with MC type methods, our method not only achieves an accuracy beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computations. Effectively, we have developed a numerical calculus of PDFs.

GMRA has many other applications that we are currently pursuing. In particular, since many classical operators of mathematical physics can be accurately approximated via a linear combination of Gaussians, using GMRA allows explicit evaluation of integrals which can lead to important advances in a number of applications.
This is a joint work with Lucas Monz´on and Ignas Satkauskas.

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