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

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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?

JohnRos
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Interesante pregunta. Sprott (2000) dice "La probabilidad fiducial no ha sido ampliamente aceptada. Esto se debe principalmente al hecho de que su uso irrestricto produce contradicciones. Por lo tanto, es importante subrayar los supuestos sobre los cuales el uso anterior de la probabilidad fiducial ..." pp. 77. También da referencias sobre estas contradicciones, como Barnard (1987) . Este documento se ha utilizado para argumentar que Fisher "vio la luz" y se convirtió en bayesiano.
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Pensé que había leído que Fisher nunca completó su argumento fiducial o al menos nunca lo hizo bastante consistente. Un artículo de AMS de 1964 de Dempster dice que "se concluye que la forma general del argumento fiducial es atractiva, pero que muchas de las restricciones impuestas por Fisher son incómodas o ambiguas y, tal vez, deberían ser reemplazadas".
Wayne
@Wayne: La referencia de Dempster es reveladora. Gracias.
JohnRos
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When I was a graduate student at Stanford (giving away my age) some 35 years ago we had a seminar course "On Rereading Fisher." The title of the seminar came from a paper by that title that had been published a year or so earlier (maybe by Jimmie Savage). Anyway each student that was taking the seminar for a grade had to read one of Fisher's papers and report on it. Mine was on one about the famous Behrens-Fisher problem. The fiducial argument was prominent in that paper. My memory of the paper and the class is not sharp being that it was 35 years ago.
Michael R. Chernick
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Fisher murió en la década de 1960 en Australia. Esto fue mucho antes de que me convirtiera en estadístico. Creo que Fisher pensó que la teoría fiducial estaba completa. Creo que otros estadísticos le hicieron agujeros y luchó por defenderlo. Pero si has leído a Fisher, sabes que era obstinado y siempre convencido de que tenía razón (lo que debe ser en ese momento). No he visto el artículo de Barnard, pero dudo que Fisher haya renunciado alguna vez a la inferencia fiducial y también dudo que se haya convertido en bayesiano.
Michael R. Chernick

Respuestas:

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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 θ given x as fid(θ|x)F/θ where F(x,θ) denotes a one-parameter cumulative distribution function for the random variable X at x with parameter θ. 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 σ in the Behrens-Fisher problem. So a fiducial distribution is like a posterior distribution for the parameter θ given the observed data x. But it is constructed without the inclusion of a prior distribution on θ.

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.

Burch, B. D. and Iyer, H. K. (1997). Exact confidence intervals for a variance ratio (or heri- tability) in a mixed linear model. Biometrics 53, 1318-1333.

Burdick, R. K., Borror, C. M. and Montgomery, D. C. (2005a). Design and Analysis of Gauge R&R Studies. ASA-SIAM Series on Statistics and Applied Probability. Philadelphia, PA: Society for Industrial and Applied Mathematics.

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

Ghosh, J. K. (1994). Higher Order Assymptotics. NSF-CBMS Regional Conference Series. Hay- ward: Institute of Mathematical Statistics.

Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer Series in Statistics. Springer-Verlag, New York.

Glagovskiy, Y. S. (2006). Construction of Fiducial Confidence Intervals For the Mixture of Cauchy and Normal Distributions. Master’s thesis, Department of Statistics, Colorado State University.

Grundy, P. M. (1956). Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter. J. Roy. Statist. Soc. Ser. B 18, 217-221.

GUM (1995). Guide to the Expression of Uncertainty in Measurement. International Organiza- tion for Standardization (ISO), Geneva, Switzerland.

Hamada, M. and Weerahandi, S. (2000). Measurement system assessment via generalized infer- ence. J. Quality Tech. 32, 241-253.

Hannig, J. (1996). On conditional distributions as limits of martingales. Mgr. thesis, (in czech), Charles University, Prague, Czech Republic.

Hannig, J., E, L., Abdel-Karim, A. and Iyer, H. K. (2006a) Simultaneous fiducial generalized confidence intervals for ratios of means of lognormal distributions. Austral. J. Statist. 35, 261-269.

Hannig, J., Iyer, H. K. and Patterson, P. (2006b) Fiducial generalized confidence intervals. J. Amer. Statist. Assoc. 101, 254-269.

Hannig, J. and Lee, T. C. M. (2007). Generalized fiducial inference for wavelet regression. Tech. rep., Colorado State University.

Iyer, H. K. and Patterson, P. (2002). A recipe for constructing generalized pivotal quantities and generalized confidence intervals. Tech. Rep. 2002/10, Department of Statistics, Colorado State University.

Iyer, H. K., Wang, C. M. J. and Mathew, T. (2004). Models and confidence intervals for true values in interlaboratory trials. J. Amer. Statist. Assoc. 99, 1060-1071.

Jeffreys, H. (1940). Note on the Behrens-Fisher formula. Ann. Eugenics 10, 48-51.

Jeffreys, H. (1961). Theory of Probability. Clarendon Press, Oxford, third edn.

Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics. Springer Series in Statistics. New York: Springer-Verlag, second edn.

Liao, C. T. and Iyer, H. K. (2004). A tolerance interval for the normal distribution with several variance components. Statist. Sinica 14, 217-229.

Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. Roy. Statist. Soc. Ser. B 20, 102-107.

McNally, R. J., Iyer, H. K. and Mathew, T. (2003). Tests for individual and population bioe- quivalence based on generalized p-values. Statistics in Medicine 22, 31-53.

Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of Statistics. McGraw-Hill, third edn.

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Salome, D. (1998). Staristical Inference via Fiducial Methods. Ph.D. thesis, University of Gronin- gen. 544 JAN HANNIG

Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. John Wiley & Sons, New York.

Stevens, W. L. (1950). Fiducial limits of the parameter of a discontinuous distribution. Biometrika 37, 117-129.

Tsui, K.-W. and Weerahandi, S. (1989). Generalized p-values in significance testing of hypothe- ses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84, 602-607.

Wang, C. M. and Iyer, H. K. (2005). Propagation of uncertainties in measurements using gen- eralized inference. Metrologia 42, 145-153.

Wang, C. M. and Iyer, H. K. (2006a). A generalized confidence interval for a measurand in the presence of type-A and type-B uncertainties. Measurement 39, 856–863. Wang, C. M. and Iyer, H. K. (2006b). Uncertainty analysis for vector measurands using fiducial inference. Metrologia 43, 486-494.

Weerahandi, S. (1993). Generalized confidence intervals. J. Amer. Statist. Assoc. 88, 899-905.

Weerahandi, S. (2004). Generalized Inference in Repeated Measures. Wiley, Hoboken, NJ.

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Yeo, I.-K. and Johnson, R. A. (2001). A uniform strong law of large numbers for U-statistics with application to transforming to near symmetry. Statist. Probab. Lett. 51, 63-69.

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: [email protected] (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

Michael R. Chernick
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You have to wait until the expiration date...
jbowman
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@MichaelChernick: I was hoping for an explanation of the argument and it's flaws. I do not feel current answers, while very useful, are complete.
JohnRos
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@JohnRos: I added to my answer which I think makes mine complete. In general i feel that giving someone a specific reference that provides a complete answer is complete enough. I think that the poser of the question who is really interested in the answwer should go to the trouble of looking at the references and learn that way. We are all grown ups and we don't have to be spoon fed everything.
Michael R. Chernick
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Scroll down and you will see @hbaghishani got it
Macro
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@MichaelChernick, I don't think there's a lot to be gained by complaining about getting downvoted/not upvoted/not winning a bounty. If anything, this will probably make users less likely to pay attention to/vote on your posts in the future. It's pretty clear to me that you put more effort into your answer (although it could have benefited from improved organization) but, ultimately, choices for votes are a matter of opinion - the real answer is probably "I liked hbaghishani's answer better", why should he have to say/explain that? Also, you may look to JohnRos's comment above for answers.
Macro
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Fiducial inference sometimes interprets likelihoods as probabilities for the parameter θ. That is, M(x)L(θ|x), provided that M(x) is finite, is interpreted as a probability density function for θ in which L(θ|x) is the likelihood function of θ and M(x)=(L(θ|x)dθ)1. You can see Casella and Berger, pages 291-2, for more details.

hbaghishani
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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.

Michael R. Chernick
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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 α2 and 1α2 quantiles of a normal distribution centered at X¯ with scale σn?

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

Brani Vidakovic
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