A comparison of bayes-laplace, jeffreys, and other priors: the case of zero events

Beta distributions with both parameters equal to 0, ½, or 1 are the usual choices for “noninformative” priors for Bayesian estimation of the binomial parameter. However, as illustrated by two examples from the Bayesian literature, care needs to be taken with parameter values below 1, both for noninformative and informative priors, as such priors concentrate their mass close to 0 and/or 1 and can suppress the importance of the observed data. These examples concern the case of no successes (or failures) and illustrate the informativeness of the Jeffreys prior usually recommended as the “consensus prior.” In particular, the second example suggests that when the binomial parameter is known to be very small, an informative prior from the beta(1, b) family (b > 1) seems appropriate, while a beta(a, b) with a < 1 can be too informative. It is thus argued that sensitivity analysis of an informative prior should be based on a consensus posterior corresponding to the Bayes–Laplace prior rather than the Jeffreys prior