A Geometric Perspective on Bayesian and Generalized Fiducial Inference

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.

arXiv comment32 pages, 4 figures

References in corpus (9)

• The frontier of simulation-based inference
• Adaptive approximate Bayesian computation
• Inferential models: A framework for prior-free posterior probabilistic inference
• Overall Objective Priors
• Marginal inferential models: prior-free probabilistic inference on interest parameters
• Conditional inferential models: combining information for prior-free probabilistic inference
• Generalized fiducial inference for normal linear mixed models
• Generalized Fiducial Inference on Differentiable Manifolds
• Introduction to Generalized Fiducial Inference