Bayesian Multilevel Model Calibration for Inverse Problems Under Uncertainty with Perfect Data

Abstract

Hierarchical or multilevel modeling establishes a convenient framework for solving complex inverse problems [1,2] in the presence of uncertainty. In the last two decades it has been studied from a frequentist [3] and a Bayesian perspective [4]. We will adopt a Bayesian point of view to statistical inversion and uncertainty quantification and present a Bayesian multilevel framework that allows for inversion and optimal analysis of "perfect" or noise-free data in the presence of aleatory and epistemic types of uncertainty and in experimental situations when data is scarce or expensive to acquire. In this contribution to the annual MascotNum workshop we will discuss the abovementioned framework on the basis of an application example within the domain of aerospace engineering [5]. We will not only illustrate the very potential of Bayesian multilevel modeling as well as ways to overcome its immanent major challenges, but more importantly we will discuss the main observations, considerations and key questions that the practical problem solution [6] has given rise to.

A unified framework for multilevel uncertainty quantification in Bayesian inverse problems

“…The data ⟨ỹ i ⟩ is then only explained by uncertainty of the forward model inputs as described in Section 2.2, without being subject to prediction errors. Hereafter we will refer this scenario as to involve "perfect" data [59,60].…”

“…Rather than aiming at an unknown constant m, inference concentrates on the hyperparameters θ X that determine the variability of envisaged. An application example where inference targets both parameters of the type m and θ X , in the presence of additional nuisance parameters ⟨ζ i ⟩, can be found in [59,60].…”

“…Therefore free parameters of the algorithm have to be chosen endeavoring high posterior fidelity, i.e. the degree as to which the induced long-run distribution conforms with the true posterior[59,60].7.4. Advanced MCMC SamplersSummarized Bayesian multilevel model calibration requires an enormous number of forward model runs.Therefore in the statistical literature a wide range of advanced MCMC techniques, dedicated to posterior exploration in classical hierarchical models, have been devised.…”

Estimating aircraft drag polar using open flight surveillance data and a stochastic total energy model

“…The application of this Bayesian approach is slowly being brought to the engineering domain. For example, Nagel and Sudret (2015) outlined a multi-level system estimation application using such a hierarchical model approach. The STE model and solutions proposed in this paper are closely related to this line of research.…”

Spectral likelihood expansions for Bayesian inference

“…the prior does not integrate to one or any finite value, and models with pathologic priors such as the Cauchy distribution for which the moments are not defined [45,46]. Many hierarchical Bayesian models [47,48] are not covered by the devised problem formulation. They are either based on conditional priors, which does not allow for orthogonal polynomials, or on integrated likelihoods, which can only be evaluated subject to noise.…”