A taxonomy of estimator consistency on discrete estimation problems

Abstract

We describe a four-level hierarchy mapping both all discrete estimation problems and all estimators on these problems, such that the hierarchy describes each estimator's consistency guarantees on each problem class. We show that no estimator is consistent for all estimation problems, but that some estimators, such as Maximum A Posteriori, are consistent for the widest possible class of discrete estimation problems. For Maximum Likelihood and Approximate Maximum Likelihood estimators we show that they do not provide consistency on as wide a class, but define a sub-class of problems characterised by their consistency. Lastly, we show that some popular estimators, specifically Strict Minimum Message Length, do not provide consistency guarantees even within the sub-class.