  Approximate Bayesian computation (ABC) methods perform inference on
model-specific parameters of mechanistically motivated parametric statistical
models when evaluating likelihoods is difficult. Central to the success of ABC
methods is computationally inexpensive simulation of data sets from the
parametric model of interest. However, when simulating data sets from a model
is so computationally expensive that the posterior distribution of parameters
cannot be adequately sampled by ABC, inference is not straightforward. We
present approximate approximate Bayesian computation" (AABC), a class of
methods that extends simulation-based inference by ABC to models in which
simulating data is expensive. In AABC, we first simulate a limited number of
data sets that is computationally feasible to simulate from the parametric
model. We use these data sets as fixed background information to inform a
non-mechanistic statistical model that approximates the correct parametric
model and enables efficient simulation of a large number of data sets by
Bayesian resampling methods. We show that under mild assumptions, the posterior
distribution obtained by AABC converges to the posterior distribution obtained
by ABC, as the number of data sets simulated from the parametric model and the
sample size of the observed data set increase simultaneously. We illustrate the
performance of AABC on a population-genetic model of natural selection, as well
as on a model of the admixture history of hybrid populations.
