This paper introduces nonparametric Bayesian credibility without imposing stringent parametric assumptions on claim distributions. We suppose that a claim distribution associated with an unknown risk characteristic of a policyholder is an unknown parameter vector with infinite dimension. In this way, we incorporate the uncertainty of the functional form of the claim distribution associated with the unknown risk characteristic in calculating credibility premiums. Using the results of Ferguson (1973), formulas of the Bayesian credibility premiums are obtained. The formula for the Bayesian credibility pure premium is a linear combination of the overall mean and the sample mean of the claims. This is consistent with the result in the classical credibility theory. We perform a simulation study for the nonparametric Bayesian credibility pure premiums and compare them with the corresponding Bühlmann credibility premiums. Estimation results for the credibility premiums using Danish fire insurance loss data are presented.