The Lindley paradox: The loss of resolution in Bayesian inference

There are three principle paradigms of statistical inference: Bayesian, frequentist and information-based inference. Although these paradigms are essentially in agreement in some contexts, the Lindley paradox describes a class of problems, models of unknown dimension, where conflicting conclusions are generated by frequentist and Bayesian inference. This conflict can materially affect the scientific conclusions. Understanding the Lindley paradox, where it applies, why it occurs and how it can be avoided, is therefore essential to the understanding of statistical analysis. In this paper, we revisit the Lindley paradox in the context of a simple biophysical application. We describe how predictive and postdictive measures of model performance provide a natural framework for understanding the Lindley paradox. We then describe which methods result in optimal experimental resolution for discovery.