Fiducial Inference - Background

Background

The general approach of fiducial inference was proposed by R A Fisher. Here "fiducial" comes from the Latin for faith. Fiducial inference can be interpreted as an attempt to perform inverse probability without calling on prior probability distributions. Fiducial inference quickly attracted controversy and was never widely accepted. Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published. These counter-examples cast doubt on the coherence of "fiducial inference" as a system of statistical inference or inductive logic. Other studies showed that, where the steps of fiducial inference are said to lead to "fiducial probabilities" (or "fiducial distributions"), these probabilities lack the property of additivity, and so cannot constitute a probability measure.

The concept of fiducial inference can be outlined by comparing its treatment of the problem of interval estimation in relation to other modes of statistical inference.

• A confidence interval, in frequentist inference, with coverage probability γ has the interpretation that among all confidence intervals computed by the same method, a proportion γ will contain the true value that needs to be estimated. This has either a repeated sampling (or frequentist) interpretation, or is the probability that an interval calculated from yet-to-be-sampled data will cover the true value. However, in either case, the probability concerned is not the probability that the true value is in the particular interval that has been calculated since at that stage both the true value and the calculated are fixed and are not random.

• Credible intervals, in Bayesian inference, do allow a probability to be given for the event that an interval, once it has been calculated does include the true value, since it proceeds on the basis that a probability distribution can be associated with the state of knowledge about the true value, both before and after the sample of data has been obtained.

Fisher’s fiducial method was designed to meet perceived problems with the Bayesian approach, at a time when the frequentist approach had yet to be fully developed. Such problems related to the need to assign a prior distribution to the unknown values. The aim was to have a procedure whose results could still be given the interpretation that a probability could be assigned to whether or not a calculated interval includes the true value. The method proceeds by attempting to derive a "fiducial distribution", which is a measure of the degree of faith that can be put on any given value of the unknown parameter.

Unfortunately Fisher did not give a general definition of the fiducial method and he denied that the method could always be applied. His only examples were for a single parameter; different generalisations have been given when there are several parameters. A relatively complete presentation of the fiducial approach to inference is given by Quenouille (1958), while Williams (1959) describes the application of fiducial analysis to the calibration problem (also known as "inverse regression") in regression analysis. Further discussion of fiducial inference is given by Kendall & Stuart (1973).