The General Formulation of Loss-Based Priors for Parameter Spaces

TL;DR

This paper introduces a unified geometric framework for loss-based priors applicable to both discrete and continuous parameter spaces, extending existing methods by considering local neighborhoods around parameters.

Contribution

It proposes a neighborhood-exclusion approach that generalizes loss-based priors to continuous parameters using local geometry and Kullback--Leibler divergence.

Findings

In one dimension, the prior matches Jeffreys' prior.

In higher dimensions, it defines a family of priors based on local geometry.

Provides a geometric interpretation of objective priors beyond isotropic cases.

Abstract

Loss-based priors assign probability mass to parameter values according to the inferential loss incurred when they are excluded from the parameter space, and provide a general solution for discrete parameters. Extending this idea to continuous settings is challenging, as the exclusion of a single point induces no loss. We propose a neighbourhood-exclusion framework in which inferential loss is defined by removing a local region around each parameter value. Under standard regularity conditions, this yields a class of prior distributions driven by the local geometry of the Kullback--Leibler divergence. In one dimension, the resulting prior coincides with Jeffreys' prior, while in higher dimensions it leads to a family of priors indexed by the geometry of the exclusion region. The proposed formulation provides a unified extension of loss-based priors and offers a geometric interpretation of objective prior construction beyond isotropic settings.