Fiducial inference: Difference between revisions

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==Background==
 
The general approach of fiducial inference was proposed by [[Ronald Fisher]].<ref>Fisher, R. A. (1935) "The fiducial argument in statistical inference", ''Annals of Eugenics'', 5, 391–398. http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1935.tb02120.x/epdf</ref><ref>[http://www.hss.cmu.edu/philosophy/seidenfeld/relating%20to%20Fisher/Fisher's%20Fiducial%20Argument%20and%20Bayes%20Theorem.pdf R. A. Fisher's Fiducial Argument and Bayes' Theorem by Teddy Seidenfeld]</ref> 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 distribution]]s.<ref>Quenouille (1958), Chapter 6</ref> Fiducial inference quickly attracted controversy and was never widely accepted<ref>Neyman, Jerzy. "Note on an article by Sir Ronald Fisher." Journal of the Royal Statistical Society. Series B (Methodological) (1956): 288-294288–294.</ref>. Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published.{{Citation needed|date=June 2011}} 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]].{{Citation needed|date=June 2011}}
 
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.
Fisher designed the fiducial method 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, like the Bayesian method, whose results could still be given an inverse probability interpretation based on the actual data observed. 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 and is faithful to the data in the sense that the method uses all available information.
 
Unfortunately Fisher did not give a general definition of the fiducial method and he denied that the method could always be applied.{{Citation needed|date=June 2011}} 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 (statistics)|calibration]] problem (also known as "inverse regression") in [[regression analysis]].<ref>Williams (1959, Chapter 6)</ref> Further discussion of fiducial inference is given by Kendall & Stuart (1973).<ref name=KS>Kendall, M. G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition'', Griffin. {{ISBN|0-85264-215-6}} (Chapter 21)</ref>
 
==The fiducial distribution==
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, JGJ. G. 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|J. G. |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 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."<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>
}}</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."
 
However, fiducial inference is still being studied and its principles appear valuable for some scientific applications.<ref>Hannig, J. (2009) "Generalized fiducial inference for wavelet regression" ''[[Biometrika]]'', 96(4),847&ndash;860847–860.</ref><ref>Hannig, J. (2009) "On generalized fiducial inference", ''Statistica Sinica'', 19, 491&ndash;544491–544</ref> In the mid-2010s, the [[psychometrics|psychometrician]] [[Yang Liu (psychometrician)|Yang Liu]] developed generalized fiducial inference for models in [[item response theory]] and demonstrated favorable results compared to frequentist and Bayesian approaches. Other current work in fiducial inference is ongoing under the name of [[confidence distribution]]s.
 
{{More footnotes|date=February 2010}}