A Bayesian Multilevel Approach to Optimally Estimate Material Properties

Abstract

Bayesian multilevel modeling establishes a convenient framework to account for aleatory variability and epistemic uncertainty in inverse problems. In view of engineering applications we will place this framework into the context of classical and probabilistic inversion and demonstrate its additional potential to infer material properties as intermediate variables within a hierarchically defined Bayesian multilevel model. The objective of probabilistic inversion is to infer such probability distributions that describe the variability of model inputs across a number of experiments. Inference therefore focuses on so-called hyperparameters that parametrize population distributions. To that end intermediate or latent variables are considered nuisance and commonly marginalized out from the problem. But instead of integrating out intermediate quantities, one can just as well fade the hyperparameters from the inferential spotlight and marginalize them out. This allows for an "optimal" inference of the intermediate quantities with respect to the available prior information, structural assumptions and the acquired data. We will devise a simple academic example within the domain of mechanical engineering that will be used to study this optimal kind of inference. The system under consideration is an ensemble of simply supported beams with uncertain material properties. Over the sample of beams, Young's moduli are subject to aleatory variability, while the knowledge about the true hyperparameter values suffers from epistemic uncertainty. We will show how individual Young's moduli can be optimally estimated in a series of three-point bending experiments. It will be shown that in hierarchically defined models one can learn more about a specific specimen than what is encoded in dedicated data directly associated with that single specimen. One can indirectly utilize information from different specimens, within the same sample, in order to improve the assessment of a specific one. In this contribution we will demonstrate the potential and challenge of this learning mechanism and we will investigate the underlying flow of information.

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