  Approximate Bayesian computation (ABC) or likelihood-free inference
algorithms are used to find approximations to posterior distributions without
making explicit use of the likelihood function, depending instead on simulation
of sample data sets from the model. In this paper we show that under the
assumption of the existence of a uniform additive model error term, ABC
algorithms give exact results when sufficient summaries are used. This
interpretation allows the approximation made in many previous application
papers to be understood, and should guide the choice of metric and tolerance in
future work. ABC algorithms can be generalized by replacing the 0-1 cut-off
with an acceptance probability that varies with the distance of the simulated
data from the observed data. The acceptance density gives the distribution of
the error term, enabling the uniform error usually used to be replaced by a
general distribution. This generalization can also be applied to approximate
Markov chain Monte Carlo algorithms. In light of this work, ABC algorithms can
be seen as calibration techniques for implicit stochastic models, inferring
parameter values in light of the computer model, data, prior beliefs about the
parameter values, and any measurement or model errors.
