Bridging Bayesian, frequentist and fiducial (BFF) inferences using confidence distribution

TL;DR

This paper explores how Bayesian, frequentist, and fiducial inference methods can be unified and compared using confidence distribution theory and Monte Carlo simulations, enhancing statistical inference techniques.

Contribution

It introduces a framework that bridges Bayesian, frequentist, and fiducial inferences, enabling their comparison and integration through confidence distributions and simulation.

Findings

Unified framework for the three paradigms

Enhanced understanding of their relationships

Potential for new hybrid inference methods

Abstract

Bayesian, frequentist and fiducial (BFF) inferences are much more congruous than they have been perceived historically in the scientific community (cf., Reid and Cox 2015; Kass 2011; Efron 1998). Most practitioners are probably more familiar with the two dominant statistical inferential paradigms, Bayesian inference and frequentist inference. The third, lesser known fiducial inference paradigm was pioneered by R.A. Fisher in an attempt to define an inversion procedure for inference as an alternative to Bayes' theorem. Although each paradigm has its own strengths and limitations subject to their different philosophical underpinnings, this article intends to bridge these different inferential methodologies through the lenses of confidence distribution theory and Monte-Carlo simulation procedures. This article attempts to understand how these three distinct paradigms, Bayesian, frequentist, and fiducial inference, can be unified and compared on a foundational level, thereby increasing the range of possible techniques available to both statistical theorists and practitioners across all fields.