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 distribution

for the prior.

Now suppose that we decide to observe the number of failures y before the mth success. Evidently, there will be m successes and y failures, and the probability of any particular sequence with that number of successes and failures is . The number of such sequences is , because the y failures and m–1 of the successes can occur in any order, but the sequence must conclude with a success. It follows that

that is, that has a negative binomial distribution (see Appendix A). For such a distribution

so that

Because it follows that

so that Jeffreys’ rule implies that we should take a prior

that is, instead of

7.4.2 Conclusion

Consequently, on being told that an experiment resulted in, say, ten successes and ten failures, Jeffreys’ rule does not allow us to decide which prior to use until we know whether the experimental design involved a fixed number of trials, or waiting until a fixed number of successes, or some other method. This clearly violates the likelihood principle (cf. Lindley, 1971a, Section 12.4); insofar as they appear to include Jeffreys’ work, it is hard to see how Berger and Wolpert (1988, Section 4.1.2) come to the conclusion that ‘… use of noninformative priors, purposely not involving subjective prior opinions … is consistent with the LP [Likelihood Principle]’). Some further difficulties inherent in the notion of a uniform reference prior are discussed in Hill (1980) and in Berger and Wolpert (1988).

However, it has been argued that a reference prior should express ignorance relative to the information which can be supplied by a particular experiment; see Box and Tiao (1992, Section 1.3). In any case, provided they are used critically, reference priors can be very useful, and, of course, if there is a reasonable amount of detail, the precise form of the prior adopted will not make a great deal of difference.