Fiducial inference: Difference between revisions

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(I changed a < X / \omega to X > \omega / a, since it is X not a that is regarded as a random variable.)
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[[File:Biologist and statistician Ronald Fisher.jpg|thumb|right|200px|Biologist and statistician Ronald Fisher]]
'''Fiducial inference''' is one of a number of different types of [[statistical inference]]. These are rules, intended for general application, by which conclusions can be drawn from [[Sample (statistics)|samples]] of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of [[frequentist inference]], [[Bayesian inference]] and [[decision theory]]. However, fiducial inference is important in the [[history of statistics]] since its development led to the parallel development of concepts and tools in [[theoretical statistics]] that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it.
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 [[frequency probability|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 interval]]s, 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.
 
</ref> and, also writing to Barnard, Fisher complained that his theory seemed to have only "an asymptotic approach to intelligibility".<ref name=Z/> 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".<ref name=Z/>
 
Lindley{{Citation needed|date=August 2011}}<ref>Sharon Bertsch McGrayne (2011) The Theory That Would Not Die. p. 133 {{full citation needed|date=November 2012}}</ref> showed that fiducial probability lacked additivity, and so was not a [[probability measure]]. Cox points out<ref>Cox (2006) p. 66</ref> that the same argument applies to the so-called "[[Confidence Distribution|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.{{Citation needed|date=February 2010}}<ref name=KS/>
 
In 1978, JG Pederson wrote that "the fiducial argument has had very limited success and is now essentially dead."<ref>{{Cite journal|doi=10.2307/1402811|first=JG| last=Pederson| title=Fiducial Inference |journal=International Statistical Review | volume= 46 | year= 1978 | pages= 147–170 | mr=0514060 | issue= 2|postscript=<!--None-->|jstor=1402811 }}</ref> Davison<ref>Davison, A.C. (2001) "''Biometrika'' Centenary: Theory and general methodology" ''[[Biometrika]]'' 2001 (page 12 in the republication edited by D. M. Titterton and [[David R. Cox]])</ref> 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."
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