Summary Estimating the parameter when is known or simultaneous estimation of and of the binomial distribution based on independent observations has been considered by many authors over the last several decades. A range of estimators have been proposed, and questions regarding asymptotic and small sample properties received adequate treatment. In this paper, we provide an extensive review and a comprehensive performance comparison of the estimators from the literature. We propose a conceptually simple estimator of that uses the marginal likelihood when is integrated out by simultaneous optimisation w.r.t. and the hyperparameters. We compare the proposed estimator with various existing estimators and find its performance competitive and, in some scenarios, superior.