Evaluations of the beta probability integral by bayes and price

Summary The contribution of Bayes to statistical inference has been much discussed, whereas his evaluations of the beta probability integral have received little attention, and Price's improvements of these results have never been analysed in detail. It is the purpose of the present paper to redress this state of affairs and to show that the Bayes-Price approximation to the two-sided beta probability integral is considerably better than the normal approximation, which became popular under the influence of Laplace, although it had been stated by Price.The Bayes-Price results are obtained by approximating the skew beta density by a symmetric beta density times a factor tending to unity for n , the two functions having the same maximum and the same points of inflection. Since the symmetric beta density converges to the normal density, all the results of Laplace based on the normal distribution can be obtained as simple limits of the results of Bayes and Price. This fact was not observed either by Laplace or by Todhunter.