Education

Abstract

Most students taking laboratory science courses have had the experience of struggling with noisy measurements of basic processes. It's especially frustrating when the process is something like a falling object which has an elementary theoretical description. As our technology improves, it is easier and easier to acquire data, but the meaningful interpretation of data requires considerable care. The Education section paper in this issue explores parameter estimation in an experiment where mapping the data to known reference values with reasonable confidence is surprisingly difficult. If this sounds familiar to you, it may be because the Education section explored this issue two years ago. In this issue, our featured paper is a self-contained, extension of an Education section contribution from early 2013 by Moritz Allmaras et al. entitled “Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example.” In that paper, the authors used Bayesian inversion to distill collected data and estimate the physical parameters associated with a falling object. In particular, the authors used digitized video data of a falling object to infer gravitational acceleration, its reduced drag coefficient, and the time in which it started falling. The Bayesian inversion technique required assumptions about the probability density functions (PDFs) of the unknown parameters and the experimental noise in measurements of the falling object. The key assumptions were the functional form and width of the prior densities used in the inversion process. In the end, the mean value of gravitational acceleration was not close to the known reference value, but the authors found that the confidence intervals adequately included the reference value. In this follow-up article, “Statistics of Parameter Estimates: A Concrete Example,” authors Oscar Aguilar, Moritz Allmaras, Wolfgang Bangerth, and Luis Tenorio take this project one step further and validate the Bayesian inversion method used in the 2013 experiment. The first validation method uses the results from the inversion on a series of simulated experiments to see if the results lie in the predicted confidence interval. The investigators also explore how the form and width of the prior PDF impacts the symmetry and structure of the posterior distributions. Here, they identify considerable sensitivity between the form of the prior density and the posterior confidence intervals. Their improved credible intervals no longer include the reference value of gravitational acceleration. Finally, the authors consider alternatives to Bayesian inversion, including maximum likelihood estimates and nonlinear least squares, and show that these results are consistent with those using Bayesian inversion. In summary, the Education section paper this issue is a clear description of both a collection of statistical methods for analyzing data and the process of validating parameter estimates acquired through statistical techniques. This paper will have considerable appeal as a module for instructors and students in statistics or mathematical modeling courses working at the advanced undergraduate level. Also, it may be useful to those working with data both in laboratory science courses and research projects.