Approximate Bayesian Computation (ABC) is a useful class of methods for
Bayesian inference when the likelihood function is computationally intractable.
In practice, the basic ABC algorithm may be inefficient in the presence of
discrepancy between prior and posterior. Therefore, more elaborate methods,
such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be
used. However, the elaboration of a proposal density for MCMC is a sensitive
issue and very difficult in the ABC setting, where the likelihood is
intractable. We discuss an automatic proposal distribution useful for ABC-MCMC
algorithms. This proposal is inspired by the theory of quasi-likelihood (QL)
functions and is obtained by modelling the distribution of the summary
statistics as a function of the parameters. Essentially, given a real-valued
vector of summary statistics, we reparametrize the model by means of a
regression function of the statistics on parameters, obtained by sampling from
the original model in a pilot-run simulation study. The QL theory is well
established for a scalar parameter, and it is shown that when the conditional
variance of the summary statistic is assumed constant, the QL has a closed-form
normal density. This idea of constructing proposal distributions is extended to
non constant variance and to real-valued parameter vectors. The method is
illustrated by several examples and by an application to a real problem in
population genetics.Comment: Published at http://dx.doi.org/10.1214/14-BA921 in the Bayesian
Analysis (http://projecteuclid.org/euclid.ba) by the International Society of
Bayesian Analysis (http://bayesian.org/