Published methods for the pricing of weather derivatives are based on classical statistics, in that the predictions of the distributions of weather indices that they use are based on best estimates of model parameters. It is likely that such methods do not accurately capture the true uncertainty because they either ignore or approximate parameter and model uncertainty. In this article we derive the objective Bayesian versions of the flat-line and linear-trend models. The use of objective Bayesian statistics allows us to incorporate parameter uncertainty into the predictive distribution in a simple way. The result is a change in the shape of the predictive distribution, but no change in the predicted mean and variance, when compared with comparable classical statistical methods. We also derive a Bayesian version of the damped linear-trend model, which, in a limited sense, also incorporates model uncertainty.