  The development of statistical methods and numerical algorithms for model
choice is vital to many real-world applications. In practice, the ABC approach
can be instrumental for sequential model design; however, the theoretical basis
of its use has been questioned. We present a measure-theoretic framework for
using the ABC error towards model choice and describe how easily existing
rejection, Metropolis-Hastings and sequential importance sampling ABC
algorithms are extended for the purpose of model checking. Considering a panel
of applications from evolutionary biology to dynamic systems, we discuss the
choice of summaries which differs from standard ABC approaches. The methods and
algorithms presented here may provide the workhorse machinery for an
exploratory approach to ABC model choice, particularly as the application of
standard Bayesian tools can prove impossible.
