A Geometric Perspective on Bayesian and Generalized Fiducial Inference

TL;DR

This paper introduces a geometric framework to understand Bayesian and generalized fiducial inference, revealing their connection via differentiable manifolds and enabling new sampling methods.

Contribution

It provides a novel geometric characterization of post-data inference, unifying Bayesian and GFI on the same manifold, and develops manifold MCMC algorithms for sampling.

Findings

Bayesian posteriors and GFDs lie on the same differentiable manifold.

The geometric approach facilitates sampling from posteriors and GFDs.

Application demonstrated with a repeated-measures ANOVA example.

Abstract

Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When a priori knowledge about parameters is available, post-data inference can be conveniently made from Bayesian posteriors. In the absence of prior information, we may still rely on objective Bayes or generalized fiducial inference (GFI). Inspired by approximate Bayesian computation, we propose a novel characterization of post-data inference with the aid of differential geometry. Under suitable smoothness conditions, we establish that Bayesian posteriors and generalized fiducial distributions (GFDs) can be respectively characterized by absolutely continuous distributions supported on the same differentiable manifold: The manifold is uniquely determined by the observed data and the data generating equation of the fitted model. Our geometric analysis not only sheds light on the connection and distinction between Bayesian inference and GFI, but also allows us to sample from posteriors and GFDs using manifold Markov chain Monte Carlo algorithms. A repeated-measures analysis of variance example is presented to illustrate the sampling procedure.