Past Event: Oden Institute Seminar

Posterior error control in Bayesian Inverse Problems

J Andrés Christen, Centro de Investigación en Matemáticas, Guanajuato, México

Abstract

In the Bayesian analysis of Inverse Problems most relevant cases the forward maps are defined in terms of a system of (O, P)DE's that involve numerical solvers. These then lead to a numerical/approximate posterior distribution. Recently several results have been published on the regularity conditions required on such numerical methods to ensure converge of the numerical to the theoretical posterior. However, more practical guidelines are needed. I present some recent results that, by using Bayes Factors and in a finite dimensional setting, one can see that the numerical posterior tends to the theoretical posterior in the same order as the numerical solver used in the forward map. Moreover, when error estimates are available for the solver we can use a bound on this errors, proven to lead to basically error free posteriors. That is, given that we are observing noisy data, we may tolerate an amount (relative to the data noise) of numerical error in the solver, and end up with a basically error free posterior. In this talk I will show these results, present some examples in ODEs and PDEs and comment on the generalizations to the infinite dimensional setting. Bio Dr J Andres Christen has a BSc in mathematics from Universidad Nacional Autonoma de Mexico (1989, UNAM, Mexico city) and a PhD in mathematics from the University of Nottingham (1994, Nottingham, UK). His expertise is in Bayesian Statistics. Dr Christen has been working on the field for more than 25 years in applied as well as theoretical aspects of the discipline. His areas of application include ecology, paleoecolgy, environmental change, bioinformatics, among others. Moreover, recently at CIMAT he has helped to create a group on the study of inverse problems using Bayesian statistics (Bayesian UQ). In particular, Dr Christen and the group at CIMAT are working on the analysis of complex biological systems, epidemiology, sound wave scattering and other applications in physics, as well as some theoretical aspects involved in the practice of Bayesian UQ. Dr Christen holds a tenure position at CIMAT (part of the national network of research centers of CONACyT) since 2003 and is a Investigador Nacional, Sistema Nacional de Investigadores level III. He is a member of the International Society for Bayesian Analysis (ISBA), participated in the Scientific Committee of ISBA 2009{2012 and was the chair of the Local Organizing Committee, 2014 ISBA World Congress. (Personal home page www.cimat.mx/~jac/. See also uq.cimat.mx for more details on the UQ group at CIMAT.)