Bayesian Probabilistic Numerical Methods: Neyman Seminar
Seminar: Neyman Seminar | January 29 | 4-5 p.m. | 1011 Evans Hall
Tim Sullivan, Freie Universität Berlin and Zuse Institute Berlin
Numerical computation --- such as numerical solution of a PDE, or quadrature --- can modelled as a statistical inverse problem in its own right. In particular, we can apply the Bayesian approach to inversion, so that a posterior distribution is induced over the object of interest (e.g. the PDE's solution) by conditioning a prior distribution on the same finite information that would be used in a classical numerical method, thereby restricting attention to a meaningful subclass of probabilistic numerical methods, one that turns out to be distinct from classical average-case analysis and information-based complexity. The main technical consideration here is that the data are non-random and thus the standard Bayes' theorem does not hold. General conditions will be presented under which numerical methods based upon such Bayesian probabilistic foundations are well-posed, and a sequential Monte Carlo method will be shown to provide consistent estimation of the posterior. The paradigm is extended to computational pipelines, through which a distributional quantification of numerical error can be propagated. A sufficient condition is presented for when such propagation yields a globally coherent Bayesian interpretation, based on a novel class of probabilistic graphical models designed to represent a computational work-flow. The concepts are illustrated through explicit numerical experiments involving both linear and non-linear models, and various directions for further research, development, and applications will be discussed.
Berkeley, CA 94720, 5106422781