¿Cuál es el argumento fiducial y por qué no ha sido aceptado?

Una de las contribuciones tardías de RA Fisher fueron los intervalos fiduciales y los argumentos de principios fiduciales . Sin embargo, este enfoque no es tan popular como los argumentos de principios bayesianos o frecuentistas. ¿Cuál es el argumento fiducial y por qué no se ha aceptado?

I am surprised that you don't consider us authorities. Here is a good reference: Encyclopedia of Biostatistics, Volume 2, page 1526; article titled "Fisher, Ronald Aylmer." Starting at the bottom of the first column on the page and going through most of the second column the authors Joan Fisher Box (R. A. Fisher's daughter) and A. W. F. Edwards write

Fisher introduced the the fiducial argument in 1930 [11].... Controversy arose immediately. fisher had proposed the fiducial argument as an alternative to the Bayesian argument of inverse probability, which he condemned when no objective prior probability could be stated.

Continúan discutiendo los debates con Jeffreys y Neyman (particularmente Neyman en intervalos de confianza). La teoría de Neyman-Pearson de la prueba de hipótesis y los intervalos de confianza surgieron en la década de 1930 después del artículo de Fisher. Siguió una oración clave.

Las dificultades posteriores con el argumento fiducial surgieron en casos de estimación multivariante debido a la falta de uniformidad de los pivotales.

En el mismo volumen de la Enciclopedia de Bioestadística hay un artículo pp. 1510-1515 titulado "Probabilidad Fiducial" por Teddy Seidenfeld que cubre el método en detalle y compara los intervalos fiduciales con los intervalos de confianza. Para citar el último párrafo de ese artículo,

En una conferencia de 1963 sobre probabilidad fiducial, Savage escribió "El objetivo de la probabilidad fiducial ... parece ser lo que yo llamo hacer la tortilla bayesiana sin romper los huevos bayesianos". En ese sentido, la probabilidad fiducial es imposible. Al igual que con muchas grandes contribuciones intelectuales, lo que tiene un valor duradero es lo que aprendemos tratando de comprender las ideas de Fisher sobre la probabilidad fiducial. (Ver Edwards [4] para mucho más sobre este tema). Su solución al problema de Behrens-Fisher, por ejemplo, fue un tratamiento brillante de los parámetros molestos usando el teorema de Bayes. En este sentido, "... el argumento fiducial es 'aprender de Fisher' [36, p.926]. Así interpretado, ciertamente sigue siendo una valiosa adición al saber estadístico.

Creo que en estas últimas oraciones Edwards está tratando de poner una luz favorable sobre Fisher a pesar de que su teoría fue desacreditada. Estoy seguro de que puede encontrar una gran cantidad de información al respecto revisando estos artículos de la enciclopedia y otros similares en otros documentos estadísticos, así como artículos biográficos y libros sobre Fisher.

Algunas otras referencias

Box, J. Fisher (1978). "TA Fisher: la vida de un científico". Wiley, Nueva York Fisher, RA (1930) Probabilidad inversa. Actas de la Sociedad Filosófica de Cambridge. 26, 528-535.

Bennett, editor de JH (1990) Análisis e inferencia estadística: correspondencia seleccionada de RA Fisher. Clarendon Press, Oxford.

Edwards, AWF (1995). La inferencia fiducial y la teoría fundamental de la selección natural. Biometría 51,799-809.

Savage LJ (1963) Discusión. Boletín del Instituto Internacional de Estadística 40, 925-927.

Seidenfeld, T. (1979). "Problemas filosóficos de inferencia estadística" Reidel, Dordrecht. Seidenfeld, T. (1992). El argumento fiducial de RA Fisher y el teorema de Bayes. Ciencia Estadística 7, 358-368.

Tukey, JW (1957). Algunos ejemplos con relevancia fiducial. Anales de Estadística Matemática 28, 687-695.

Zabell, SL (1992). RA Fisher y el argumento fiducial. Ciencia estadística 7, 369-387.

El concepto es difícil de entender porque el pescador lo siguió cambiando como dijo Seidenfeld en su artículo en la Enciclopedia de Bioestadística

Following the 1930 publication, during the remaining 32 years of his life, through two books and numerous articles , Fisher steadfastly held to the idea captured in (1), and the reasoning leading to it which we may call'fiducial inverse inference' then there is little wonder that Fisher caused such puzzles with his novel idea

Equation (1) that Seidenfeld refers to is the fiducial distribution of the parameter $\theta$ given $x$ as $\text{fid}(\theta|x) \propto \partial F/\partial \theta$ where $F(x,\theta)$ denotes a one-parameter cumulative distribution function for the random variable $X$ at $x$ with parameter $\theta$. At least this was Fisher's initial definition. Later it got extended to multiple parameters and that is where the trouble began with the nuisance parameter $\sigma$ in the Behrens-Fisher problem. So a fiducial distribution is like a posterior distribution for the parameter $\theta$ given the observed data $x$. But it is constructed without the inclusion of a prior distribution on $\theta$.

I went to some trouble getting all this but it is not hard to find. We are really not needed to answer questions like this. A Google search with key words "fiducial inference" would likely show everything I found and a whole lot more.

I did a Google search and found that a UNC Professor Jan Hannig has generalized fiducial inference in an attempt to improve it. A Google search yields a number of his recent papers and a powerpoint presentation. I am going to copy and paste the last two slides from his presentation below:

Concluding Remarks

Generalized fiducial distributions lead often to attractive solution with asymptotically correct frequentist coverage.

Many simulation studies show that generalized fiducial solutions have very good small sample properties.

Current popularity of generalized inference in some applied circles suggests that if computers were available 70 years ago, fiducial inference might not have been rejected.

Quotes

Zabell (1992) “Fiducial inference stands as R. A. Fisher’s one great failure.” Efron (1998) “Maybe Fisher’s biggest blunder will become a big hit in the 21st century! "

Just to add more references, here is the reference list I have taken from Hannig's 2009 Statistics Sinica paper. Pardon the repetition but I think this will be helpful.

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Burdick, R. K., Park, Y.-J., Montgomery, D. C. and Borror, C. M. (2005b). Confidence intervals for misclassification rates in a gauge R&R study. J. Quality Tech. 37, 294-303.

Cai, T. T. (2005). One-sided confidence intervals in discrete distributions. J. Statist. Plann. Inference 131, 63-88.

Casella, G. and Berger, R. L. (2002). Statistical Inference. Wadsworth and Brooks/Cole Ad- vanced Books and Software, Pacific Grove, CA, second edn.

Daniels, L., Burdick, R. K. and Quiroz, J. (2005). Confidence Intervals in a Gauge R&R Study with Fixed Operators. J. Quality Tech. 37, 179-185.

Dawid, A. P. and Stone, M. (1982). The functional-model basis of fiducial inference. Ann. Statist. 10, 1054-1074. With discussions by G. A. Barnard and by D. A. S. Fraser, and a reply by the authors.

Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference. J. Roy. Statist. Soc. Ser. B 35, 189-233. With discussion by D. J. Bartholomew, A. D. McLaren, D. V. Lindley, Bradley Efron, J. Dickey, G. N. Wilkinson, A. P.Dempster, D. V. Hinkley, M. R. Novick, Seymour Geisser, D. A. S. Fraser and A. Zellner, and a reply by A. P. Dawid, M. Stone, and J. V. Zidek.

Dempster, A. P. (1966). New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Statist. 37, 355-374.

Dempster, A. P. (1968). A generalization of Bayesian inference. (With discussion). J. Roy. Statist. Soc. B 30, 205-247.

Dempster, A. P. (2008). The Dempster-Shafer calculus for statisticians. International Journal of Approximate Reasoning 48, 365-377.

E, L., Hannig, J. and Iyer, H. K. (2008). Fiducial intervals for variance components in an unbalanced two-component normal mixed linear model. J. Amer. Statist. Assoc. 103, 854- 865.

Efron, B. (1998). R. A. Fisher in the 21st century. Statist. Sci. 13, 95-122. With comments and a rejoinder by the author.

Fisher, R. A. (1930). Inverse probability. Proceedings of the Cambridge Philosophical Society xxvi, 528-535.

Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to unknown parameters. Proceedings of the Royal Society of London A 139, 343-348.

Fisher, R. A. (1935a). The fiducial argument in statistical inference. Ann. Eugenics VI, 91-98.

Fisher, R. A. (1935b). The logic of inductive inference. J. Roy. Statist. Soc. B 98, 29-82.

Fraser, D. A. S. (1961). On fiducial inference. Ann. Math. Statist. 32, 661-676.

Fraser, D. A. S. (1966). Structural probability and a generalization. Biometrika 53, 1–9.

Fraser, D. A. S. (1968). The Structure of Inference. John Wiley & Sons, New York-London- Sydney.

Fraser, D. A. S. (2006). Fiducial inference. In The New Palgrave Dictionary of Economics (Edited by S. Durlauf and L. Blume). Palgrave Macmillan, 2nd edition. ON GENERALIZED FIDUCIAL INFERENCE 543

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Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci. 7, 369-387. Department of Statistics and Operations Research, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3260, U.S.A. E-mail: hannig@unc.edu (Received November 2006; accepted December 2007)

The article i got this from is Statistica Sinica 19 (2009), 491-544 ON GENERALIZED FIDUCIAL INFERENCE∗ Jan Hannig The University of North Carolina at Chapel Hill

Fiducial inference sometimes interprets likelihoods as probabilities for the parameter $\theta$. That is, $M(x)L(\theta|x)$, provided that $M(x)$ is finite, is interpreted as a probability density function for $\theta$ in which $L(\theta|x)$ is the likelihood function of $\theta$ and $M(x)=(\int_{-\infty}^{\infty}L(\theta|x)d\theta)^{-1}$. You can see Casella and Berger, pages 291-2, for more details.

Just to add to what is said, there was controversy between Fisher and Neyman about significance testing and interval estimation. Neyman defined confidence intervals while Fisher introduced fiducial intervals. They argued differently about their construction but the constructed intervals were usually the same. So the difference in the definitions was largely ignored until it was discovered that they differed when dealing with the Behrens-Fisher problem. Fisher argued adamantly for the fiducial appraoch but inspite of his brillance and his strong advocation of the method, there appeared to be flaws and since the statistical community considers it discredited it is not commonly discussed or used. The Bayesian and frequentist approaches to inference are the two that remain.

In a large undergraduate class of engineering intro stats at Georgia Tech, when discussing confidence intervals for the population mean with variance known, one student asked me (in the language of MATLAB): "Can I calculate the interval as > norminv([alpha/2,1-alpha/2], barX, sigma/sqrt(n))?" In translation: could he take $\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$ quantiles of a normal distribution centered at $\bar X$ with scale $\frac\sigma{\sqrt{n}}$?

I said – of course YES, pleasantly surprised that he naturally arrived to the concept fiducial distribution.