TY - JOUR

T1 - Statistics of Parameter Estimates: A Concrete Example

AU - Aguilar, Oscar

AU - Allmaras, Moritz

AU - Bangerth, Wolfgang

AU - Tenorio, Luis

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The work of this author was partially supported by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).The work of this author was partially supported by NSF grant DMS-0914987.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2015/1

Y1 - 2015/1

N2 - © 2015 Society for Industrial and Applied Mathematics. Most mathematical models include parameters that need to be determined from measurements. The estimated values of these parameters and their uncertainties depend on assumptions made about noise levels, models, or prior knowledge. But what can we say about the validity of such estimates, and the influence of these assumptions? This paper is concerned with methods to address these questions, and for didactic purposes it is written in the context of a concrete nonlinear parameter estimation problem. We will use the results of a physical experiment conducted by Allmaras et al. at Texas A&M University [M. Allmaras et al., SIAM Rev., 55 (2013), pp. 149-167] to illustrate the importance of validation procedures for statistical parameter estimation. We describe statistical methods and data analysis tools to check the choices of likelihood and prior distributions, and provide examples of how to compare Bayesian results with those obtained by non-Bayesian methods based on different types of assumptions. We explain how different statistical methods can be used in complementary ways to improve the understanding of parameter estimates and their uncertainties.

AB - © 2015 Society for Industrial and Applied Mathematics. Most mathematical models include parameters that need to be determined from measurements. The estimated values of these parameters and their uncertainties depend on assumptions made about noise levels, models, or prior knowledge. But what can we say about the validity of such estimates, and the influence of these assumptions? This paper is concerned with methods to address these questions, and for didactic purposes it is written in the context of a concrete nonlinear parameter estimation problem. We will use the results of a physical experiment conducted by Allmaras et al. at Texas A&M University [M. Allmaras et al., SIAM Rev., 55 (2013), pp. 149-167] to illustrate the importance of validation procedures for statistical parameter estimation. We describe statistical methods and data analysis tools to check the choices of likelihood and prior distributions, and provide examples of how to compare Bayesian results with those obtained by non-Bayesian methods based on different types of assumptions. We explain how different statistical methods can be used in complementary ways to improve the understanding of parameter estimates and their uncertainties.

UR - http://hdl.handle.net/10754/599732

UR - http://epubs.siam.org/doi/10.1137/130929230

UR - http://www.scopus.com/inward/record.url?scp=84923095214&partnerID=8YFLogxK

U2 - 10.1137/130929230

DO - 10.1137/130929230

M3 - Article

VL - 57

SP - 131

EP - 149

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 1

ER -