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RESEARCH PRODUCT

Rejoinder on: Natural Induction: An Objective Bayesian Approach

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Giron and Moreno. We certainly agree with Professors Giron and Moreno on the interest in sensitivity of any Bayesian result to changes in the prior. That said, we also consider of considerable pragmatic importance to be able to single out a unique, particular prior which may reasonably be proposed as the reference prior for the problem under study, in the sense that the corresponding posterior of the quantity of interest could be routinely used in practice when no useful prior information is available or acceptable. This is precisely what we have tried to do for the twin problems of the rule of succession and the law of natural induction. The discussants consider the limiting binomial version of the Law of Natural Induction, and focused on the version that can be stated in the language of hypothesis testing involving H0 ≡ {p = 1}. They then noted that a popular objective Bayesian approach to hypothesis testing is to use intrinsic priors on the alternative, which tend to be more concentrated about the null value than the Be(p | 1/2, 1/2) prior we use. The notion is that, if a problem is posed as that of testing a ‘privileged’ null hypothesis, then realistic alternatives will tend to be close to the null value, and the prior distribution – even in supposedly objective procedures – should reflect this. Thus a strong case can be made for use of intrinsic priors in that setting. The natural induction problem, however, is not a problem with a privileged null hypothesis in this sense; there is no a-priori notion that values of p near 1 are more believable than other values. Hence we would argue that, for the natural induction problem, the analysis we propose is the preferred objective procedure.