Fiducial inference (Fisher, 1930) produces a fiducial distribution for parameters without having to specify a prior probability distribution.
The technique was acrimoniously debated in the statistical community for many years, practically disappeared after Fisher’s death, and resulted in confusion as to what the mysterious fiducial argument even is. It is perhaps best known as Fisher’s great failure (Zabell, 1992).
“[Fiducial inference] is an attempt to make the Bayesian omelette without breaking the Bayesian eggs.” ~ Savage (1962)
A Brief History
Fisher developed Fiducial inference as an objective, inferential alternative to “subjective” Bayesian reasoning. The first iteration of the method was barely indistinguishable from Neyman’s (1935) unconditional confidence interval approach. Over the next two decades, Fisher worked to clarify his reasoning, but his shift in thinking led to more confusion. Ultimately, Fisher concluded that his new intuitions about the method were fundamentally incorrect. In 1963, Buehler & Fedderson showed that one of Fisher’s crucial arguments was false.
Problems with Fiducial Inference
Fiducial inference It isn’t widely used, due to a number of conceptual and operational difficulties. For example, “…its unrestricted use produces contradictions” (Sprott, 2000, p.77). Other problems include the fact that discrete data and continuous parameters play interchanging roles, which means it can’t be used at all for discrete data.
References:
Buehler, R. & Fedderson, A. Note on conditional property of Student’s t. Ann. Math. Statist. 34. 1098-1100.
Fisher, R. A. (1930). Inverse Probability. Proceedings of the Cambridge Philosophical Society 26 528-535.
Neyman, J. (1935). On the problem of confidence limits. Annals of Mathematical Statistics 6, 111–116.
Savage, L. (1962). Discussion of Birnbaum, A, On the foundations of statistical inference (with discussion). J/ Amer. Statist. Assoc. 57 269-306.
Sprott, D. (2000). Statistical Inference in Science. Springer Series in Statistics.
Welsh, A. (1996). Aspects of Fiducial Inference. Wiley-Interscience.
Zabell, S. R. A. Fisher and the Fiducial Argument. Statistical Science. Vol 7, No. 3, 369-387.