The prior can generally only be understood in the context of the likelihood

TL;DR

This paper discusses the conceptual challenge of choosing priors in Bayesian analysis, emphasizing that priors should be understood independently of the likelihood, and proposes a framework integrating prior choice with the full Bayesian process.

Contribution

It clarifies the conceptual tension in prior modeling and proposes a framework that contextualizes prior choice within the entire Bayesian analysis process.

Findings

Highlights the paradox in prior modeling methods

Proposes a new perspective on prior and likelihood independence

Integrates prior choice into inference, prediction, and evaluation

Abstract

A key sticking point of Bayesian analysis is the choice of prior distribution, and there is a vast literature on potential defaults including uniform priors, Jeffreys' priors, reference priors, maximum entropy priors, and weakly informative priors. These methods, however, often manifest a key conceptual tension in prior modeling: a model encoding true prior information should be chosen without reference to the model of the measurement process, but almost all common prior modeling techniques are implicitly motivated by a reference likelihood. In this paper we resolve this apparent paradox by placing the choice of prior into the context of the entire Bayesian analysis, from inference to prediction to model evaluation.