Bayesian inference without point estimates

Abstract

It is conventional to apply Bayes' formula only to point estimates of the prior probabilities. This convention is unnecessarily restrictive. The analyst may prefer to estimate that the priors belong to some set of probability vectors. Set estimates allow the non-paradoxical expression of ignorance and support rigorous inference on such everyday assertions as one event is more likely than another or that an event usually occurs. Bayes' formula can revise set estimates, often at little computational cost beyond that needed for point priors. Set estimates can also inform statistical decisions, although disagreement exists about what decision methods are best.