Following a X validated question, I read a 1994 paper by Phil Dawid on the selection paradoxes in Bayesian statistics, which first sounded like another version of the stopping rule paradox. And upon reading, less so. As described above, the issue stands with drawing inference on the index and value, (i⁰,μ⁰), of the largest mean of a sample of Normal rvs. What I find surprising in Phil’s presentation is that the Bayesian analysis does not sound that Bayesian. If given the whole sample, a Bayesian approach should produce a posterior distribution on (i⁰,μ⁰), rather than follow estimation steps, from estimating i⁰ to estimating the associated mean. And if needed, estimators should come from the definition of a particular loss function. When, instead, given the largest point in the sample, and only that point, its distribution changes, so I am fairly bemused by the statement that no adjustment is needed.
The prior modelling is also rather surprising in that the priors on the means should be joint rather than a product of independent Normals, since these means are compared and hence comparable. For instance a hierarchical prior seems more appropriate, with location and scale to be estimated from the whole data. Creating a connection between the means… A relevant objection to the use of independent improper priors is that the maximum mean μ⁰ then does not have a well-defined measure. However, I do not think a criticism of some priors versus other is a relevant attack on this “paradox”.
