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| '''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.
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| ==Background==
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| The general approach of fiducial inference was proposed by [[Ronald Fisher|R A Fisher]].{{Citation needed|date=June 2011}} 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.{{Citation needed|date=June 2011}} 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}}
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| 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.
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| *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.
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| *[[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.
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| 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.
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| 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>
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| ==The fiducial distribution==
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| Fisher required the existence of a [[sufficient statistic]] for the fiducial method to apply. Suppose there is a single sufficient statistic for a single parameter. That is, suppose that the [[conditional distribution]] of the data given the statistic does not depend on the value of the parameter. For example suppose that ''n'' independent observations are uniformly distributed on the interval <math>[0,\omega]</math>. The maximum, ''X'', of the ''n'' observations is a [[sufficient statistic]] for ω. If only ''X'' is recorded and the values of the remaining observations are forgotten, these remaining observations are equally likely to have had any values in the interval <math>[0,X]</math>. This statement does not depend on the value of ω. Then ''X'' contains all the available information about ω and the other observations could have given no further information.
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| The [[cumulative distribution function]] of ''X'' is
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| :<math>F(x) = P(X \leq x) = P\left(\mathrm{all\ observations} \leq x\right) = \left(\frac{x}{\omega}\right)^n .</math> | |
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| Probability statements about ''X''/ω may be made. For example, given α, a value of ''a'' can be chosen with 0 < ''a'' < 1 such that
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| :<math>P\left(a < \frac{X}{\omega}\right) = 1-a^n = \alpha.</math>
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| Thus
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| :<math>a = (1-\alpha)^{\frac{1}{n}} .</math>
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| Then Fisher says{{Citation needed|date=October 2010}} that this statement may be inverted into the form
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| :<math>P\left(\omega < \frac{X}{a}\right) = \alpha .</math>
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| In this latter statement, ω is now regarded as a [[random variable]] and ''X'' is fixed, whereas previously it was the other way round. This distribution of ω is the ''fiducial distribution'' which may be used to form fiducial intervals.
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| The calculation is identical to the [[pivotal quantity|pivotal method]] for finding a confidence interval, but the interpretation is different. In fact older books use the terms ''confidence interval'' and ''fiducial interval'' interchangeably.{{Citation needed|date=October 2010}} Notice that the fiducial distribution is uniquely defined when a single sufficient statistic exists.
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| The pivotal method is based on a random variable that is a function of both the observations and the parameters but whose distribution does not depend on the parameter. Such random variables are called [[pivotal quantity|pivotal quantities]]. By using these, probability statements about the observations and parameters may be made in which the probabilities do not depend on the parameters and these may be inverted by solving for the parameters in much the same way as in the example above. However, this is only equivalent to the fiducial method if the pivotal quantity is uniquely defined based on a sufficient statistic.
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| A fiducial interval could be taken to be just a different name for a confidence interval and give it the fiducial interpretation. But the definition might not then be unique.{{Citation needed|date=October 2010}} Fisher would have denied that this interpretation is correct: for him, the fiducial distribution had to be defined uniquely and it had to use all the information in the sample.{{Citation needed|date=October 2010}}
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| ==Status of the approach==
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| 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.
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| 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,<ref name=Z>{{cite journal|doi=10.1214/ss/1177011233|title=R. A. Fisher and Fiducial Argument
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| |first=S. L.
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| |last=Zabell <!-- |authorlink=Sandy L. Zabell -->
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| |journal=Statistical Science
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| |volume=7
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| |issue=3
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| |date=Aug 1992
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| |pages=369–387
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| |jstor=2246073
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| }} (page 381)
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| </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/>
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| Lindley{{Citation needed|date=August 2011}}<ref>Sharon Bertsch McGrayne (2011) The Theory That Would Not Die. p. 133 {{full|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/>
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| 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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| However, fiducial inference is still being studied<ref>Hannig, J. (2009) "Generalized fiducial inference for wavelet regression" ''[[Biometrika]]'', 96(4),847–860.</ref><ref>Hannig, J. (2009) "On generalized fiducial inference", ''Statistica Sinica'', 19, 491–544</ref> and other current work is ongoing under the name of [[confidence distribution]]s.
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| {{More footnotes|date=February 2010}}
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *[[David R. Cox|Cox, D. R.]] (2006). ''Principles of Statistical Inference'', CUP. ISBN 0-521-68567-2.
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| *{{cite book |last=Fisher |first=R A |coauthors= |title=Statistical Methods and Scientific Inference|year=1956 |publisher=Hafner |location=New York |isbn=0-02-844740-9}}
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| * [[Ronald A. Fisher|Fisher, Ronald]] "Statistical methods and scientific induction" ''Journal of the Royal Statistical Society, Series B'' 17 (1955), 69—78. (criticism of statistical theories of [[Jerzy Neyman]] and [[Abraham Wald]] from a fiducial perspective)
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| * {{cite journal|
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| title=Note on an Article by Sir Ronald Fisher
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| |authorlink=Jerzy Neyman
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| |first=Jerzy
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| |last=Neyman
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| |journal=[[Journal of the Royal Statistical Society, Series B]]
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| |volume=18
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| |issue=2
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| |year=1956
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| |pages=288–294
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| |jstor=2983716
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| }} (reply to Fisher 1955, which diagnoses a fallacy of "fiducial inference")
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| *{{cite book |editor=Tukey, J W, ed.|last=|first= |coauthors= |title=R.A. Fisher's Contributions to Mathematical Statistics|year=1950|publisher=Wiley |location=New York }}
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| *Quenouille, M.H. (1958) ''Fundamentals of Statistical Reasoning''. Griffin, London
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| *Williams, E.J. (1959) ''Regression Analysis'', Wiley {{LCCN|59011815}}
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| *Young, G.A., Smith, R.L. (2005) ''Essentials of Statistical Inference'', CUP. ISBN 0-521-83971-8
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| * Fraser, D.A.S. (1961) "The fiducial method and invariance." ''Biometrika'', '''48''', 261-80.
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| * Fraser, D.A.S. (1961) "On fiducial inference." ''Annals of Mathematical Statistics,'' '''32''', 661-676.
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| {{DEFAULTSORT:Fiducial Inference}}
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| [[Category:Statistical theory]]
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| [[Category:Statistical inference]]
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Hi there, I am Yoshiko Villareal but I by no means really liked that title. Meter reading is where my main income arrives from but soon I'll be on my personal. Climbing is what love doing. Some time in the past he chose to reside in Kansas.
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