3.2 Reference prior for the binomial likelihood
3.2.1 Bayes’ postulate
The Rev. Thomas Bayes himself in Bayes (1763) put forward arguments in favour of a uniform prior
(which, unlike the choice of a prior uniform over , is a proper density in that it integrates to unity) as the appropriate one to use when we are ‘completely ignorant’. This choice of prior has long been known as Bayes’ postulate, as distinct from his theorem. The same prior was used by Laplace (1774). It is a member of the conjugate family, to wit Be(1, 1).
Bayes’ arguments are quite intricate, and still repay study. Nevertheless, he seems to have had some doubts about the validity of the postulate, and these doubts appear to have been partly responsible for the fact that his paper was not published in his lifetime, but rather communicated posthumously by his friend Richard Price.
The postulate seems intuitively reasonable, in that it seems to treat all values on a level and thus reflect the fact that you so no reason for preferring any one value to any other. However, you should not be too hasty in endorsing it because ignorance about the value of π presumably implies ignorance about the value of any function of π, and yet when the change of variable rule is used a uniform prior for π will not usually imply a uniform ...