What does "fiducial" mean (in the context of statistics)?

Question

When I Google for

"fisher" "fiducial"

...I sure get a lot of hits, but all the ones I've followed are utterly beyond my comprehension.

All these hits do seem to have one thing in common: they are all written for dyed-in-the-wool statisticians, people thoroughly steeped in the theory, practice, history, and lore of statistics. (Hence, none of these accounts bothers to explain or even illustrate what Fisher meant by "fiducial" without resorting to oceans of jargon and/or passing the buck to some classic or other of the mathematical statistics literature.)

Well, I don't belong to the select intended audience that could benefit for what I've found on the subject, and maybe this explains why every one of my attempts to understand what Fisher meant by "fiducial" has crashed against a wall of incomprehensible gibberish.

Does anyone know of an attempt to explain to someone who is not a professional statistician what Fisher meant by "fiducial"?

P.S. I realize that Fisher was a bit of a moving target when it came to pinning down what he meant by "fiducial", but I figure the term must have some "constant core" of meaning, otherwise it could not function (as it clearly does) as terminology that is generally understood within the field.

Answer 1

The fiducial argument is to interpret likelihood as a probability.
Edit: not exactly, see the answer of @Sextus for more details.

Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up on the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

Answer 2

Several well-known statisticians try to rekindle an interest in Fisher's fiducial argument. Bradley Efron: (I cannot copy even small quotes from google books), the topic is also treated in Bradley Efron 2. He says something to the effect of (not a direct quote): Fiducial inference, sometimes considered Fisher's largest error, can be Fisher largest hit for the future. So there are people thinking that Fiducial ideas will come back.

A complete book devoted to the topic (by some of my former professors) is Schweder, T. & Hjort, N. L. (2016). Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions. Cambridge University Press.

They propose to change terminology from "fiducial distribution" to "confidence distribution". I even at some point tried to make a new tag here confidence-distribution. But somebody mistakenly made that a tag synonym for confidence-interval. Grrrr (If made a synonym, it should be to fiducial.)

Answer 3

In the answer from gui11aume it is stated that

The fiducial argument is to interpret likelihood as a probability.

However, there is a subtle difference between the fiducial distribution and the likelihood function. If $F(\hat\theta; \theta)$ is the cumulative distribution function of some parameter estimate $\hat\theta$ given the true parameter $\theta$ then the fiducial/confidence distribution is $\frac{d}{d\theta}F(\hat\theta,\theta)$ whereas the likelihood function is $\frac{d}{d\hat\theta}F(\hat\theta,\theta)$.*

The fiducial argument is to inverse the interpretation of the probability statement described by the cumulative distribution function. An example is described in this question The basic logic of constructing a confidence interval

The way of constructing the confidence interval in that question, by using the cumulative distribution function, is equivalent to constructing a fiducial distribution.

So this specific** type of confidence interval and related confidence distribution is what is meant by the fiducial distribution and fiducial probability. And, it is much more around then we think; it is only not named fiducial interval and overtaken by the more general confidence interval. Another reason why 'fiducial' is less popular is because of the difference in the philosophical interpretation and attempts to have a more Bayesian interpretation than frequentist interpretation. The fiducial intervals were supposed to contain the parameter 95% of the time but they only do so from a specific point of view: Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

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* For some types of cases like a rate parameter or a location parameter, the fiducial distribution coincides with the likelihood distribution because in those cases $\frac{d}{d\theta}F(\hat\theta,\theta) \propto \frac{d}{d\hat\theta}F(\hat\theta,\theta)$. But the fiducial distribution does not transform in the same way when we re-express the parameters (it scales more like a probability distribution function). For instance, the fiducial distribution for estimating $\sigma$ in a normal distribution is not the same as the likelihood function.

** Note 'specific'. The fiducial distribution is a confidence distribution, but not every confidence distribution is a fiducial distribution.