Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example


Allmaras, Moritz
Bangerth, Wolfgang
Linhart, Jean Marie
Polanco, Javier
Wang, Fang
Wang, Kainan
Webster, Jennifer
Zedler, Sarah

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All mathematical models of real-world phenomena contain parameters that need to be estimated from measurements, either for realistic predictions or simply to understand the characteristics of the model. Bayesian statistics provides a framework for parameter estimation in which uncertainties about models and measurements are translated into uncertainties in estimates of parameters. This paper provides a simple, step-by-step example-starting from a physical experiment and going through all of the mathematics-to explain the use of Bayesian techniques for estimating the coefficients of gravity and air friction in the equations describing a falling body. In the experiment we dropped an object from a known height and recorded the free fall using a video camera. The video recording was analyzed frame by frame to obtain the distance the body had fallen as a function of time, including measures of uncertainty in our data that we describe as probability densities. We explain the decisions behind the various choices of probability distributions and relate them to observed phenomena. Our measured data are then combined with a mathematical model of a falling body to obtain probability densities on the space of parameters we seek to estimate. We interpret these results and discuss sources of errors in our estimation procedure. © 2013 Society for Industrial and Applied Mathematics.

Allmaras M, Bangerth W, Linhart JM, Polanco J, Wang F, et al. (2013) Estimating Parameters in Physical Models through Bayesian Inversion: A Complete Example. SIAM Review 55: 149–167. Available:

The work of the first, third, fifth, and eighth authors was supported by award KUS-C1-016-04 from the KingAbdullah University of Science and Technology. The work of the seventh author was supported byU.S. Department of Homeland Security grant 2008-DN-077-ARI001-02.The work of this author was supported by NSF award DMS-0604778, U.S. Department of Energy grant DE-FG07-07ID14767, U.S. Department of HomelandSecurity grant 2008-DN-077-ARI001-02, award KUS-C1-016-04 from the King Abdullah Universityof Science and Technology, and an Alfred P. Sloan Research Fellowship.

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