Construction of Objective Bayesian Prior from Bertrand’s Paradox and the Principle of Indifference

The Principle of Indifference, which may be naïvely interpreted as the requirement to assign the same probability to different outcomes of a probabilistic event, is applied in Objective Bayesian analysis to form prior distributions (priors). However, though it may be desired that such priors are truly “objective”, they usually are not. This paper compares a number of usual objective priors - uniform, invariant, reference, and maximum entropy priors and examines them from an epistemological perspective to find what premises are implied if these objective priors are taken as an implementation of the Principle of Indifference that achieves complete objectivity in the resulting statistical analysis procedure. Then, given an conventional ignorance or lack-of-information interpretation for objectivity, it is found that these priors are indeed not completely “objective”. It may be possible to obtain a weaker, or more general, a priori analysis for ignorance such that there can be conceptually completely objective priors.