On the Flatland paradox and limiting arguments
Pierre Druilhet · Communication in Statistics- Theory and Methods · 2017
We revisit the Flatland paradox proposed by Stone (1976 Stone, M. 1976. Strong inconsistency from uniform priors. Journal of the American Statistical Association 71 (353):114–25.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which is an example of non conglomerability. The main novelty in the analysis of the paradox is to consider marginal versus conditional models rather than proper versus improper priors. We show that in the first model a prior distribution should be considered as a probability measure, whereas, in the second one, a prior distribution should be considered in the projective space of measures. This induces two different kinds of limiting arguments which are useful to understand the paradox. We also show that the choice of a flat prior is not adapted to the structure of the parameter space and we consider an improper prior based on reference priors with nuisance parameters for which the Bayesian analysis matches the intuitive reasoning.