Bayesian Statistics: An Introduction, 4th Edition

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

or

(as a check, this density does integrate to unity). Now it has been argued that if we ‘know nothing’ about θ then ...