  In the presence of modeling errors, the mainstream Bayesian methods seldom
give a realistic account of uncertainties as they commonly underestimate the
inherent variability of parameters. This problem is not due to any
misconception in the Bayesian framework since it is absolutely robust with
respect to the modeling assumptions and the observed data. Rather, this issue
has deep roots in users' inability to develop an appropriate class of
probabilistic models. This paper bridges this significant gap, introducing a
novel Bayesian hierarchical setting, which breaks time-history vibrational
responses into several segments so as to capture and identify the variability
of inferred parameters over multiple segments. Since computation of the
posterior distributions in hierarchical models is expensive and cumbersome,
novel marginalization strategies, asymptotic approximations, and maximum a
posteriori estimations are proposed and outlined under a computational
algorithm aiming to handle both uncertainty quantification and propagation
tasks. For the first time, the connection between the ensemble covariance
matrix and hyper distribution parameters is characterized through approximate
estimations. Experimental and numerical examples are employed to illustrate the
efficacy and efficiency of the proposed method. It is observed that, when the
segments correspond to various system conditions and input characteristics, the
proposed method delivers robust parametric uncertainties with respect to
unknown phenomena such as ambient conditions, input characteristics, and
environmental factors.
