  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.
