The General Formulation of Loss-Based Priors for Parameter Spaces

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.