  Scientists often express their understanding of the world through a
computationally demanding simulation program. Analyzing the posterior
distribution of the parameters given observations (the inverse problem) can be
extremely challenging. The Approximate Bayesian Computation (ABC) framework is
the standard statistical tool to handle these likelihood free problems, but
they require a very large number of simulations. In this work we develop two
new ABC sampling algorithms that significantly reduce the number of simulations
necessary for posterior inference. Both algorithms use confidence estimates for
the accept probability in the Metropolis Hastings step to adaptively choose the
number of necessary simulations. Our GPS-ABC algorithm stores the information
obtained from every simulation in a Gaussian process which acts as a surrogate
function for the simulated statistics. Experiments on a challenging realistic
biological problem illustrate the potential of these algorithms.
