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.

The Savage-Dickey ratio is known as a specialised representation of the Bayes
factor (O'Hagan and Forster, 2004) that allows for a functional plugging
approximation of this quantity. We demonstrate here that it is a generic
approximation method instead of an identity imposing constraints on the prior
distributions, while incidentally clarifying the measure-theoretic bases of the
method. We provide furthermore a general framework to produce a converging
approximation of the Bayes factor that is unrelated with the earlier approach
of Verdinelli and Wasserman (1995).