2.5 Locally uniform priors
2.5.1 Bayes’ postulate
We have already said that it seems useful to have a reference prior to aid public discourse in situations where prior opinions differ or are not strong. A prior which does not change very much over the region in which the likelihood is appreciable and does not take very large values outside that region is said to be locally uniform. For such a prior
so that on normalizing the posterior must equal the standardized likelihood.
Bayes himself appears to have thought that, at least in the case where θ is an unknown probability between 0 and 1, the situation where we ‘know nothing’ should be represented by taking a uniform prior and this is sometimes known as Bayes’ postulate (as distinct from his theorem).
However, it should be noted that if, for example
then on writing
we have according to the usual change-of-variable rule
(as a check, this density does integrate to unity). Now it has been argued that if we ‘know nothing’ about θ then ...