The formal definition of reference priors

TL;DR

This paper provides a rigorous, general definition of reference priors in Bayesian inference, enabling explicit derivation of these priors under weak conditions for improved objective Bayesian analysis.

Contribution

It introduces a formal, universally applicable definition of reference priors and derives explicit expressions for them under minimal regularity assumptions.

Findings

Explicit formula for reference priors under weak regularity conditions

Enables analytical and numerical derivation of reference priors

Strengthens the theoretical foundation of objective Bayesian inference

Abstract

Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.