On the Flatland Paradox

We revisit the flatland paradox proposed by \cite{ston1976} which is an example of non-conglomerability. The aim of the paper is to show that the improperness of the prior is not directly involved in the inconsistency. First, we 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 parameter for which the Bayesian analysis matches the intuitive reasoning. Then, we propose an analysis by considering the flat prior as limit of proper uniform priors. In order to use limiting arguments, we must make a distinction between two different Bayesian paradigms. The first one is related to the marginal model whereas the second one is related to the conditional model. For the latter approach, we show that the inconsistency remains even with proper priors provided that we reconsider the interpretation of prior distributions.