On the Flatland Paradox

TL;DR

This paper analyzes the flatland paradox, demonstrating that prior improperness isn't the root cause of inconsistency, and explores different Bayesian paradigms and priors to reconcile intuitive reasoning with formal analysis.

Contribution

It clarifies the role of prior choice in the flatland paradox and distinguishes between Bayesian paradigms, proposing a refined interpretation of priors to resolve inconsistencies.

Findings

Improper priors are not the main cause of the paradox.

Reference priors align Bayesian analysis with intuitive reasoning.

Inconsistency persists under certain Bayesian interpretations even with proper priors.

Abstract

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.