  Bayesian inference is often used in cosmology and astrophysics to derive
constraints on model parameters from observations. This approach relies on the
ability to compute the likelihood of the data given a choice of model
parameters. In many practical situations, the likelihood function may however
be unavailable or intractable due to non-gaussian errors, non-linear
measurements processes, or complex data formats such as catalogs and maps. In
these cases, the simulation of mock data sets can often be made through forward
modeling. We discuss how Approximate Bayesian Computation (ABC) can be used in
these cases to derive an approximation to the posterior constraints using
simulated data sets. This technique relies on the sampling of the parameter
set, a distance metric to quantify the difference between the observation and
the simulations and summary statistics to compress the information in the data.
We first review the principles of ABC and discuss its implementation using a
Population Monte-Carlo (PMC) algorithm and the Mahalanobis distance metric. We
test the performance of the implementation using a Gaussian toy model. We then
apply the ABC technique to the practical case of the calibration of image
simulations for wide field cosmological surveys. We find that the ABC analysis
is able to provide reliable parameter constraints for this problem and is
therefore a promising technique for other applications in cosmology and
astrophysics. Our implementation of the ABC PMC method is made available via a
public code release.
