# Fiducial distribution

< List of probability distributions < Fiducial distribution

A fiducial distribution in statistics is, loosely speaking, another name for a confidence distribution. The term originated with Fisher [1], who gave a general method for computing real-valued confidence limits before the formal concept of a confidence interval existed [2].

## Fisher’s fiducial distribution

Fisher put forth the idea of a fiducial distribution without having to specify a prior probability distribution.

The idea is to assume F(x, θ) is a parametric cumulative distribution. In addition., assume a “pivotal variable” μ follows a uniform distribution U(0, 1), so that

μ = F (x, θ).

If each value x, F( x, θ). is monotonic in θ, this equation will have a unique solution

θ = θ (μ, x)

for each μ ∈ (0, 1).

Fisher defined the fiducial distribution of θ (assuming no prior probabilities) as the distribution of θ, implied by θ = θ (μ, x), when x is fixed and μ is uniformly distributed.

The idea sounds relatively simple, but Fisher’s work on fiducial inference (which derives the fiducial distribution) generated intense debate in the literature. Some aspects of Fisher’s approach failed, such as applying the properties of the distribution for multi-parameter problems. In addition, Fisher’s description of pivotal variables was seen by many as confusing and restrictive [3]. This may be the reason Fisher’s work disappeared into the annals of history, earning the moniker “Fisher’s great failure” [4].

“[Fiducial inference] is an attempt to make the Bayesian omelette without breaking the Bayesian eggs.” ~ Savage (1962)

Savage [5]

That said, some authors have revived Fisher’s work in recent years under the label of generalized inference, a tool for deriving statistical procedures when frequentist methods are inadequate [6]. The main idea of generalized inference is to transfer randomness from data to the parameter space using an inverse of a data-generating equation without using Bayes’ theorem. The resulting distribution can then be used for inference [7].

## References

[1] Fisher, R. A. (1930). “Inverse probability. Proc. Camb. phil. Soc. 26, 528– 35.

[2]McCullagh, P. (2008). The fiducial method: What is it? Retrieved April 18, 2023 from: http://www.stat.uchicago.edu/~pmcc/seminars/UNC/fiducial.pdf

[3] Yager, R. (2008). Classic Works of the Dempster-Shafer Theory of Belief Functions. Springer.

[4] Savage, L. (1962). Discussion of Birnbaum, A, On the foundations of statistical inference (with discussion). J/ Amer. Statist. Assoc. 57 269-306.

[5] Zabell, S. R. A. Fisher and the Fiducial Argument. Statistical Science. Vol 7, No. 3, 369-387.

[6] Hannig, J. On Generalized Fiducial Inference.

[7] Jan Hannig, Hari Iyer, Randy C. S. Lai & Thomas C. M. Lee (2016)
Generalized Fiducial Inference: A Review and New Results, Journal of the American Statistical Association, 111:515, 1346-1361, DOI: 10.1080/01621459.2016.1165102

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