Hierarchical Bayes for the Explicit Estimation of Model Prediction Errors

An extensive research effort is dedicated to Bayesian estimation methods for analyzing the empirical behaviour of structures. State-of-the-art structural identification methods currently quantify model uncertainties by estimating hyper-parameters for the prediction-error prior. This paper exposes that this uncertainty quantification procedure does not fully recognize the epistemic nature of model prediction errors, because their posterior probability density function (PDF) is not explicitly estimated and their interaction with model parameters are not considered. This paper presents a Hierarchical Bayes formulation for estimating the joint posterior PDF of model parameters and prediction errors. This Hierarchical Bayes approach allows capturing the dependencies between unknown model parameters and unknown prediction errors; it offers a more accurate picture of the structural behaviour than when estimating the prior hyper-parameters alone. The application of this method to large-scale structures requires an adequate model the for the prediction-error prior, which remains a case-specific challenge. An extensive research effort is dedicated to Bayesian estimation methods for analyzing the empirical behaviour of structures. One of the goals of structural identification is to infer model parameters from on-site observations. Another is to employ the updated knowledge about parameters to make better predictions for unobserved quantities. Since the work of Beck and Katafygiotis (1998) in the field of structural identification, Bayesian estimation has been increasing in popularity for the tasks of inferring the properties of structures (Zhang et al. (2011); Papadimitriou et al. (2001); Au and Zhang (2016)), and for detecting damages in the context of structural health monitoring (Yuen et al. (2006); Simoen et al. (2012)). More recently the focus has shifted to Hierarchical Bayes formulations (Behmanesh et al. (2015); Huang et al. (2017); Nagel and Sudret (2016)), where Bayesian estimation is employed to identify the joint posterior probability density function (PDF) for model parameters and for the hyper-parameters of the predictionerrors prior PDF. A prediction error is defined as the discrepancy between a model prediction and the unknown true system’s response. One common aspect of current Hierarchical Bayes formulations found in the literature is that prediction errors are quantified through the estimation of the hyperparameters of the prediction-error prior PDF. This approach is limited because prediction errors are not random; they are unknown deterministic values. Similar Bayesian estimation approaches were proposed by Brynjarsdóttir and O’Hagan (2014) and by Ling et al. (2014), where the posterior PDF for prediction errors are explicitly estimated. The field of structural identification can build on these approaches for improving the way model parameters and prediction errors are estimated. For this task, two aspects remain to be addressed: (1) the formalization of the approach in a hierarchical manner and (2) the explicit separation of prediction errors into its prior and posterior estimates. Authors such as Simoen et al. (2013a,b) and Goulet et al. (2014) have already demonstrated that model simplifications introduce dependencies in