Our main interest is the prediction of future events for recurrent event processes. We intend to build a Bayesian model, based on sound philosophical principles. We consider here a homogeneous Poisson process for which there is no subjective prior information. This leads us to use the maximum entropy principle to choose the prior rather than the reference prior, that is the Jeffreys prior. The maximum entropy prior is obtained by maximizing the entropy subject to the constraint that the first two moments equal the empirical ones. This method is tested as to its effectiveness for prediction measured by the coverage probability and the length of prediction interval and its goodness-of-fit measured by the Kullback-Lieber criterion and a discrepancy measure. The estimator obtained by this method is compared to the estimators obtained using the Jeffreys prior as well as the one using the more popular conjugate prior, that is, the gamma prior. The method is also illustrated with an example concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment.