Status of The Approach
After its formulation by Fisher, fiducial inference quickly attracted controversy and was never widely accepted. Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published.
Fisher admitted that "fiducial inference" had problems. Fisher wrote to George A. Barnard that he was "not clear in the head" about one problem on fiducial inference, and, also writing to Barnard, Fisher complained that his theory seemed to have only "an asymptotic approach to intelligibility". Later Fisher confessed that "I don't understand yet what fiducial probability does. We shall have to live with it a long time before we know what it's doing for us. But it should not be ignored just because we don't yet have a clear interpretation".
Lindley showed that fiducial probability lacked additivity, and so was not a probability measure. Cox points out that the same argument applies to the so-called "confidence distribution" associated with confidence intervals, so the conclusion to be drawn from this is moot. Fisher sketched "proofs" of results using fiducial probability. When the conclusions of Fisher's fiducial arguments are not false, many have been shown to also follow from Bayesian inference.
In 1978, JG Pederson wrote that "the fiducial argument has had very limited success and is now essentially dead." Davison wrote "A few subsequent attempts have been made to resurrect fiducialism, but it now seems largely of historical importance, particularly in view of its restricted range of applicability when set alongside models of current interest."
However, fiducial inference is still being studied and other current work is ongoing under the name of confidence distributions.
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