Bayesian Inference in Models with Euclidean Structures

Abstract

When a parameter space is endowed with a Euclidean structure, there is a Euclidean measure, called the structural (or geometric) prior, which is an obvious candidate to represent ignorance concerning an unknown parameter. The rationale for using this prior is strengthened when the unknown parameter is the canonical parameter of an exponential family. Then the structural prior is the uniform prior for the canonical parameter. It is shown also that Euclidean structures can be combined in different ways with group structures. The structural (or geometric) prior is then the product of the invariant prior for the group structure and the structural (or geometric) prior for the Euclidean structure.

Key Words:

• Geometric Bayesian models
• Group models
• Inner inference
• Invariant priors