An Axiomatisation of Error Intolerant Estimation

TL;DR

This paper introduces Error Intolerance Candidates (EIC) estimators, a new class of estimators optimized for high-stakes and scientific inference scenarios, providing a Bayesian justification that unifies MAP and Wallace-Freeman estimators.

Contribution

It defines EIC estimators and proves their optimality in certain estimation scenarios, offering a new axiomatic foundation and Bayesian justification for existing estimators like MAP and WF.

Findings

EIC estimators are optimal in high-stakes estimation scenarios.

The paper unifies MAP and WF estimators under the EIC framework.

Provides a Bayesian axiomatic justification for these estimators.

Abstract

Point estimation is a fundamental statistical task. Given the wide selection of available point estimators, it is unclear, however, what, if any, would be universally-agreed theoretical reasons to generally prefer one such estimator over another. In this paper, we define a class of estimation scenarios which includes commonly-encountered problem situations such as both ``high stakes'' estimation and scientific inference, and introduce a new class of estimators, Error Intolerance Candidates (EIC) estimators, which we prove is optimal for it. EIC estimators are parameterised by an externally-given loss function. We prove, however, that even without such a loss function if one accepts a small number of incontrovertible-seeming assumptions regarding what constitutes a reasonable loss function, the optimal EIC estimator can be characterised uniquely. The optimal estimator derived in this second case is a previously-studied combination of maximum a posteriori (MAP) estimation and Wallace-Freeman (WF) estimation which has long been advocated among Minimum Message Length (MML) researchers, where it is derived as an approximation to the information-theoretic Strict MML estimator. Our results provide a novel justification for it that is purely Bayesian and requires neither approximations nor coding, placing both MAP and WF as special cases in the larger class of EIC estimators.