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

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.

Fiducial inference viewed through a possibility-theoretic inferential model lens

“…That is, instead of formulating a way to construct a data-dependent distribution Q x and hoping that it satisfies certain properties (e.g., a Bernstein-von Mises theorem), let's just define the fiducial distribution to be the best probabilistic approximation of the IM's possibilistic output. This would ensure that the fiducial distribution's credible sets are genuine confidence sets, which is what "confidence distributions" aim to achieve (e.g., Nadarajah et al 2015;Thornton and Xie 2020;Xie and Singh 2013). The challenge with the suggested strategy is actually finding the best probabilistic approximation.…”