Testing Un-Separated Hypotheses by Estimating a Distance

TL;DR

This paper introduces a Bayesian method for testing hypotheses that are not well separated by analyzing the posterior distribution of a discrepancy measure, enabling testing of complex hypotheses with asymptotic guarantees.

Contribution

It proposes a novel Bayesian approach based on posterior discrepancy measures for testing non-separated hypotheses, adaptable to complex scenarios.

Findings

Method provides asymptotic properties and calibration insights.

Achieves asymptotic frequentist optimality.

Applicable to complex hypotheses testing.

Abstract

In this paper we propose a Bayesian answer to testing problems when the hypotheses are not well separated. The idea of the method is to study the posterior distribution of a discrepancy measure between the parameter and the model we want to test for. This is shown to be equivalent to a modification of the testing loss. An advantage of this approach is that it can easily be adapted to complex hypotheses testing which are in general difficult to test for. Asymptotic properties of the test can be derived from the asymptotic behaviour of the posterior distribution of the discrepancy measure, and gives insight on possible calibrations. In addition one can derive separation rates for testing, which ensure the asymptotic frequentist optimality of our procedures.