Bayesian Statistics: An Introduction, 4th Edition

7.4 The likelihood principle and reference priors

7.4.1 The case of Bernoulli trials and its general implications

Care should be taken when using reference priors as a representation of prior ignorance. We have already seen in Section 2.4 on ‘Dominant likelihoods’ that the improper densities which often arise as reference priors should be regarded as approximations, reflecting the fact that our prior beliefs about an unknown parameter (or some function of it) are more or less uniform over a wide range. A different point to be aware of is that some ways of arriving at such priors, such as Jeffreys’ rule, depend on the experiment that is to be performed, and so on intentions. (The same objection applies, of course, to arguments based on data translated likelihoods.) Consequently, an analysis using such a prior is not in accordance with the likelihood principle.

To make this clearer, consider a sequence of independent trials, each of which results in success with probability or failure with probability (i.e. a sequence of Bernoulli trials). If we look at the number of successes x in a fixed number n of trials, so that

then, as was shown in Section 3.3, Jeffreys’ rule results in an arc-sine ...