Bayesian priors in sequential binomial design

The status of sequential analysis in Bayesian inference is revisited. The information on the experimental design, including the stopping rule, is one part of the evidence, prior to the sampling. Consequently this information must be incorporated in the prior distribution. This approach allows to relax the likelihood principle when appropriate. It is illustrated in the case of successive Binomial trials. Using Jeffreys' rule, a prior based on the Fisher information and conditional on the design characteristics is derived. The corrected Jeffreys prior, which involves a new distribution called Beta-J, extends the classical Jeffreys priors for the Binomial and Pascal sampling models to more general stopping rules. As an illustration, we show that the correction induced on the posterior is proportional to the bias induced by the stopping rule on the maximum likelihood estimator.