Estimating Parameters in Physical Models through Bayesian Inversion: AComplete Example

AbstractAll mathematical models of real-world phenomena contain parameters that needto be estimated from measurements, either for realistic predictions orsimply to understand the characteristics of the model.Bayesian statistics provides a framework for parameter estimationin which uncertainties about modelsand measurements are translated into uncertainties in estimates ofparameters. This paper provides a simple, step-by-step example---startingfrom a physical experiment and going through all of the mathematics---toexplain the use of Bayesian techniques for estimating the coefficients ofgravity and air friction in the equations describing a fallingbody.In the experiment we dropped an object from a known heightand recorded the free fall using a video camera. The video recording wasanalyzed frame by frame to obtain the distance the body had fallen as a functionof time, including measures of uncertainty in our data that we describe asprobability densities. We explain the decisions behind the various choicesof probability distributions and relate them to observed phenomena. Our measureddata are then combined with a mathematical model of a falling body to obtainprobability densities on the space of parameters we seek to estimate.We interpret these results and discuss sources of errors in our estimationprocedure.