Abstract
In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of Capistrán et al. (2016) to (a priori) expected Bayes factors (BF) and in a more general, infinite-dimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.
Highlights
“Uncertainty Quantification (UQ)” has a level of arbitrariness: all of statistics is partly concerned with quantifying uncertainty
In the inverse problems community UQ mostly refers to the theory and practice of probabilistic propagation of uncertainty and probabilistic (Bayesian) inference in the context of complex regressors involving systems of differential equations
We address in this work what we believe is an important issue of Bayesian UQ
Summary
Bayesian Uncertainty Quantification (UQ) is a recently coined term and has received considerable attention in the last ten years, by several researchers in many fields, as an alternative solution to some classical inverse problems (see Kaipio and Somersalo, 2005; Fox et al, 2013; Dashti and Stuart, 2016, for some reviews on the subject). High order solvers Capistran et al (2016) illustrate, by reducing the step size in the numerical solver, that there should exist a point at which the BF is nearly one, but for a fixed discretization threshold, α(n) (step size) greater than zero This threshold, in turn, depends on the numerical FM error compared to the data noise σ, the sample size, and other factors. The basic idea is to establish the relative merit of the numerical posterior vs the theoretical posterior using Bayesian model selection as a function of n, to keep both models with a Bayes Factor of nearly one We prove how this is possible, for finite n, resulting in a posterior distribution with negligible numerical error w.r.t the theoretical posterior.
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