Bayesian Statistics: An Introduction, 4th Edition

3.6 Reference prior for the uniform distribution

3.6.1 Lower limit of the interval fixed

If consists of independent random variables with distributions, then

where , so that the likelihood can be written in the form

after multiplying by a constant (as far as θ is concerned). Hence,

with

It follows that the likelihood is data translated, and the general argument about data translated likelihoods in Section 2.5 now suggests that we take a prior which is at least locally uniform in , that is . In terms of the parameter θ, the usual change of variable rule shows that this means

which is the same ...