  There are three principle paradigms of statistics: Bayesian, frequentist and
information-based inference. Although these paradigms are 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 identify methods which result in
optimal experimental resolution for discovery.
