This work considers the challenging problem of identifying the statistical properties of random fields from indirect observations. To this end, a Bayesian approach is introduced, whose key step is the nonparametric approximation of the likelihood function from limited information. When the likelihood function is based on the evaluation of an expensive computer code, this work also proposes a method to select iteratively new design points to reduce the uncertainties on the results that are due to the approximation of the likelihood. Two applications are finally presented to illustrate the efficiency of the proposed procedure: a first one based on analytic data, and a second one dealing with the identification of the random elasticity field of an heterogeneous microstructure.