A possibility-theoretic solution to Basu's Bayesian--frequentist via media

TL;DR

This paper proposes a possibility-theoretic approach as a middle ground between Bayesian and frequentist methods, addressing limitations of likelihood-based inference and aiming to uphold the likelihood principle.

Contribution

It introduces a novel possibility-theoretic framework to reconcile Bayesian and frequentist perspectives, overcoming issues with probabilistic inference.

Findings

Likelihood alone cannot reliably support probabilistic inference.

Possibilistic inference remains reliable within the proposed framework.

The approach offers flexibility to approximate the likelihood principle.

Abstract

Basu's via media is what he referred to as the middle road between the Bayesian and frequentist poles. He seemed skeptical that a suitable via media could be found, but I disagree. My basic claim is that the likelihood alone can't reliably support probabilistic inference, and I justify this by considering a technical trap that Basu stepped in concerning interpretation of the likelihood. While reliable probabilistic inference is out of reach, it turns out that reliable possibilistic inference is not. I lay out my proposed possibility-theoretic solution to Basu's via media and I investigate how the flexibility afforded by my imprecise-probabilistic solution can be leveraged to achieve the likelihood principle (or something close to it).