For the past couple of years Rob Salomone has been part of the University of New South Wales and the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers as a post-doctoral fellow. At the latest meeting of the New South Wales branch, held on 30th September 2020, Rob gave a very entertaining talk about Monte Carlo - the statistical version.
As is well-known, Monte Carlo is a casino city in the small Mediterranean country of Monaco. In the 1940s, researchers at Los Alamos National Laboratories in New Mexico, USA, working on atomic bomb research, borrowed the name for the idea of using sampling methods to approximate integrals. Monte Carlo methods are now a mainstay of statistical methodology. Rob described the Monte Carlo approach in general, and then recent from contributions him and his collaborators.
A recurring theme throughout Rob's presentation was "integration by darts". This involved a graphic of an image plot of a bivariate function, with dart board concentric circles superposed - and these circles being contours of a bivariate density. Throwing darts matches draws from the density, which can be used in an obvious way to approximate the expectation of the function. However, if the bivariate function has important features well away from the bull's eye then integration by darts, i.e. Monte Carlo approximation, can perform poorly. Even in this two-dimension setting the challenges were made apparent. Connections with Bayesian inference were given.
Getting into the second half of this very animated talk, Rob discussed remedies for Monte Carlo challenges such as multiplying by one and adding zero in very smart ways. One of several examples from Rob's research concernedrare events for the sum of dependent log-normal variates. He pointed to papers such as Botev, Salomone & Mackinlay (2019), Salomone, South, Drovandi and Kroese (2020) and Hodgkinson, Salomone & Roosta (2020). The last one got into Stein operators and Polish spaces - which, from appearances, involve some elegant mathematics in the name of improved Monte Carlo statistical methodology.
University of Technology Sydney