Reflections on use of Monte Carlo methods

Abstract

Monte Carlo methods have become ubiquitous in our society. Although statistical estimation methods date to earlier times, Monte Carlo (MC) methods were born in 1949 with publication of the paper by Metropolis and Ulam (1949). We address some of the many ways that MC methods have evolved over the years. Specifically, we consider electron transport calculations by MC, improvements in sampling efficiency afforded by the discrete alias sampling method, and solution of inverse problems by the Symbolic Monte Carlo (SMC) method. As a graduate student, one of us (WLD) used MC to estimate the improvement of cylindrical Geiger-Müller tube efficiency at low energies due to use of a thin coating of high atomic number material on the inside tube surface. Our methods were approximate but the results were reasonably good. We will discuss how discrete alias sampling—introduced in 1974—can improve the sampling efficiency significantly over simple table look-up. Finally, we consider an inverse problem in which the form of the governing probability density function is unknown but desired. We also briefly address matters of perspective on use of MC methods.N. Metropolis, S. Ulam, The Monte Carlo Method, 1949, J. Am. Stat. Assoc. 44, 335–341.