  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.
