  In this work we consider Bayesian inference problems with intractable
likelihood functions. We present a method to compute an approximate of the
posterior with a limited number of model simulations. The method features an
inverse Gaussian Process regression (IGPR), i.e., one from the output of a
simulation model to the input of it. Within the method, we provide an adaptive
algorithm with a tempering procedure to construct the approximations of the
marginal posterior distributions. With examples we demonstrate that IGPR has a
competitive performance compared to some commonly used algorithms, especially
in terms of statistical stability and computational efficiency, while the price
to pay is that it can only compute a weighted Gaussian approximation of the
marginal posteriors.
