Hierarchical Stochastic Model in Bayesian Inference for Engineering Applications: Theoretical Implications and Efficient Approximation

Abstract

We classify two types of Hierarchical Bayesian Model found in the literature as Hierarchical Prior Model (HPM) and Hierarchical Stochastic Model (HSM). Then, we focus on studying the theoretical implications of the HSM. Using examples of polynomial functions, we show that the HSM is capable of separating different types of uncertainties in a system and quantifying uncertainty of reduced order models under the Bayesian model class selection framework. To tackle the huge computational cost for analyzing HSM, we propose an efficient approximation scheme based on Importance Sampling and Empirical Interpolation Method. We illustrate our method using two examples -a Molecular Dynamics simulation for Krypton and a pharmacokinetic/pharmacodynamic model for cancer drug.

Hierarchical Bayesian modeling framework for model updating and robust predictions in structural dynamics using modal features

“…The available samples from the first step are subsequently utilized for computing the posterior distributions of the hyper-parameters in the second step. More details for the sampling approach can be found in [23,49]. Comparisons of the sampling approach with asymptotic techniques are presented in the next section.…”

“…In light of above, a comprehensive hierarchical Bayesian modeling (HBM) framework has been further developed in various scientific disciplines [20][21][22][23][24] to properly quantify the uncertainties within the model parameters. Specifically in the field of structural dynamics, the HBM approach was used to quantify uncertainties due to environmental variabilities [25], as well as amplitude of excitation variabilities [26].…”

Nonlinear model updating through a hierarchical Bayesian modeling framework

“…A hierarchical Bayesian modeling (HBM) framework has recently been introduced in various engineering fields [41][42][43][44][45][46][47]. In the field of structural dynamics, it was initially proposed by Behmanesh et al for structural identification based upon a mainly full simulation HBM approach [42].…”

Hierarchical Bayesian operational modal analysis: Theory and computations

“…Hierarchical Bayesian methods open up new horizons in modeling the uncertainty and have shown great promise in different scientific disciplines [36][37][38][39]. In structural dynamics, Behmanesh et al.…”