  Approximate Bayesian Computation (ABC) methods have gained in their
popularity over the last decade because they expand the horizon of Bayesian
parameter inference methods to the range of models for which only forward
simulation is available. The majority of the ABC methods rely on the choice of
a set of summary statistics to reduce the dimension of the data. However, as
has been noted in the ABC literature, the lack of convergence guarantees that
is induced by the absence of a vector of sufficient summary statistics that
assures inter-model sufficiency over the set of competing models, hinders the
use of the usual ABC methods when applied to Bayesian model selection or
assessment. In this paper, we present a novel ABC model selection procedure for
dynamical systems based on a newly appeared multi-level Markov chain Monte
Carlo method, self-regulating ABC-SubSim, and a hierarchical state-space
formulation of dynamic models. We show that this formulation makes it possible
to independently approximate the model evidence required for assessing the
posterior probability of each of the competing models. We also show that
ABC-SubSim not only provides an estimate of the model evidence as a simple
by-product but also it gives the posterior probability of each model as a
function of the tolerance level, which allows the ABC model choices made in
previous studies to be understood. We illustrate the performance of the
proposed framework for ABC model updating and model class selection by applying
it to two problems in Bayesian system identification: a single
degree-of-freedom bilinear hysteretic oscillator and a three-story shear
building with Masing hysteresis, both of which are subject to a seismic
excitation.
