The General Formulation of Loss-Based Priors for Parameter Spaces

2026-04-21

AI Summary

This work addresses the challenge of effectively extending loss-based priors from discrete to continuous parameter spaces. By introducing a neighborhood exclusion framework, it defines the inferential loss incurred by removing a local neighborhood of a parameter and leverages the local geometric structure of the Kullback–Leibler divergence to construct, under regularity conditions, a family of priors that depend on the shape of the excluded region. This approach provides a unified geometric interpretation of loss-driven priors in continuous settings: it naturally recovers Jeffreys’ prior in one dimension and yields novel prior families in higher dimensions, whose forms are determined by the geometry of the excluded neighborhoods. Consequently, the study broadens both the theoretical foundation and the construction methodology for objective priors.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• loss-based priors
• continuous parameter spaces
• inferential loss
• objective prior
• Kullback-Leibler divergence

Innovation

• loss-based priors
• neighbourhood-exclusion
• Kullback–Leibler geometry
• objective Bayesian inference
• Jeffreys' prior