Objective priors for the bivariate normal model

TL;DR

This paper explores various objective Bayesian priors for the bivariate normal model, examining their properties, optimality criteria, and surprising results such as priors that produce exact frequentist inferences for key parameters.

Contribution

It provides a comprehensive analysis of objective priors for the bivariate normal, offering recommendations and revealing novel priors that achieve exact frequentist inference.

Findings

Objective priors vary in properties and suitability.

Some priors yield exact frequentist inferences for parameters.

Surprising results include priors that simplify inference for correlation.

Abstract

Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investigation, a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, including the correlation coefficient.