Home
> fiducial inference, reviews > The generalized fiducial distribution: A kinder, more objective posterior?

## The generalized fiducial distribution: A kinder, more objective posterior?

1 June 2017

**MR3561954**

Hannig, Jan; Iyer, Hari; Lai, Randy C. S.; Lee, Thomas C. M.

Generalized fiducial inference: a review and new results. (English summary)

62A01 (62F99 62G05 62J05)

Generalized fiducial inference: a review and new results. (English summary)

*J. Amer. Statist. Assoc.*111 (2016), no. 515, 1346–1361.62A01 (62F99 62G05 62J05)

This review article introduces generalized fiducial inference, the flavor of fiducial statistics developed by the authors and their collaborators since the beginning of the millennium. This research program has been driven by a vision of fiducial distributions as posterior distributions untainted by the subjectivity seen in prior distributions.

Other approaches to fiducial inference bring subjectivity more to the forefront. For example, G. N. Wilkinson had highlighted the incoherence of fiducial distributions formulated in a more Fisherian flavor [J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326]. More recently, R. J. Bowater [AStA Adv. Stat. Anal. 101 (2017), no. 2, 177–197] endorsed an explicitly subjective interpretation of fiducial probability. For the place of generalized fiducial inference in the context of other fiducial approaches, see [D. L. Sonderegger and J. Hannig, in Contemporary developments in statistical theory, 155–189, Springer Proc. Math. Stat., 68, Springer, Cham, 2014; MR3149921] and the papers it {MR3149921} cites.

Other approaches to fiducial inference bring subjectivity more to the forefront. For example, G. N. Wilkinson had highlighted the incoherence of fiducial distributions formulated in a more Fisherian flavor [J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326]. More recently, R. J. Bowater [AStA Adv. Stat. Anal. 101 (2017), no. 2, 177–197] endorsed an explicitly subjective interpretation of fiducial probability. For the place of generalized fiducial inference in the context of other fiducial approaches, see [D. L. Sonderegger and J. Hannig, in Contemporary developments in statistical theory, 155–189, Springer Proc. Math. Stat., 68, Springer, Cham, 2014; MR3149921] and the papers it {MR3149921} cites.

In addition to providing an inspiring exposition of generalized fiducial inference, the authors report these new contributions:

- A weak-limit definition of a generalized fiducial distribution.
- Sufficient conditions for a generalized fiducial distribution to have asymptotic frequentist coverage.
- Novel formulas for computing a generalized fiducial distribution and a fiducial probability of a model.

The fiducial probability of a model is applicable to both model selection and model averaging. A seemingly different fiducial method of averaging statistical models was independently proposed by D. R. Bickel [“A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models”, preprint, Fac. Sci. Math. Stat., Univ. Ottawa, 2015].

Reviewed by David R. Bickel

**References**

- Barnard, G. A. (1995), “Pivotal Models and the Fiducial Argument,” International Statistical Reviews, 63, 309–323. [1346]
- Bayarri, M. J., Berger, J. O., Forte, A., and García-Donato, G. (2012), “Criteria for Bayesian Model Choice With Application to Variable Selection,” The Annals of Statistics, 40, 1550–1577. [1347] MR3015035
- Beaumont, M. A., Zhang, W., and Balding, D. J. (2002), “Approximate Bayesian Computation in Population Genetics,” Genetics, 162, 2025–2035. [1349]
- Berger, J. O. (1992), “On the Development of Reference Priors” (with discussion), Bayesian Statistics, 4, 35–60. [1347] MR1380269
- Berger, J. O., Bernardo, J. M., and Sun, D. (2009), “The Formal Definition of Reference Priors,” The Annals of Statistics, 37, 905–938. [1347, 1350, 1351] MR2502655
- Berger, J. O., Bernardo, J. M., and Sun, D. (2012), “Objective Priors for Discrete Parameter Spaces,” Journal of the American Statistical Association, 107, 636–648. [1347] MR2980073
- Berger, J. O., and Pericchi, L. R. (1996), “The Intrinsic Bayes Factor for Model Selection and Prediction,” Journal of the American Statistical Association, 91, 109–122. [1354] MR1394065
- Berger, J. O., and Pericchi, L. R. (2001), “Objective Bayesian Methods for Model Selection: Introduction and Comparison,” in Model Selection (IMS Lecture Notes Monogr. Ser., Vol. 38), ed. P. Lahiri, Beachwood, OH: Inst. Math. Statist., pp. 135–207. [1354] MR2000753
- Berger, J. O., and Sun, D. (2008), “Objective Priors for the Bivariate Normal Model,” The Annals of Statistics, 36, 963–982. [1347] MR2396821
- Birnbaum, A. (1961), “On the Foundations of Statistical Inference: Binary Experiments,” The Annals of Mathematical Statistics, 32, 414–435. [1348] MR0126307
- Birnbaum, A. (1962), “On the Foundations of Statistical Inference,” Journal of the American Statistical Association, 57, 269–326. [1350] MR0138176
- Blom, G. (1976), “Some Properties of Incomplete U-Statistics,” Biometrika, 63, 573–580. [1358] MR0474582
- Candes, E., and Tao, T. (2007), “The Dantzig Selector: Statistical Estimation When p is Much Larger Than n,” The Annals of Statistics, 35, 2313–2351. [1358] MR2382644
- Casella, G., and Berger, R. L. (2002), Statistical Inference (2nd ed.), Pacific Grove, CA: Wadsworth and Brooks/Cole Advanced Books and Software. [1348, 1356] MR1051420
- Chen, J., and Chen, Z. (2008), “Extended Bayesian Information Criteria for Model Selection With Large Model Spaces,” Biometrika, 95, 759–771. [1355] MR2443189
- Chiang, A. K. L. (2001), “A Simple General Method for Constructing Confidence Intervals for Functions of Variance Components,” Technometrics, 43, 356–367. [1347] MR1943189
- Cisewski, J., and Hannig, J. (2012), “Generalized Fiducial Inference for Normal Linear Mixed Models,” The Annals of Statistics, 40, 2102–2127. [1347, 1348, 1357, 1358] MR3059078
- Dawid, A. P., and Stone, M. (1982), “The Functional-Model Basis of Fiducial Inference,” The Annals of Statistics, 10, 1054–1074. [1346] MR0673643
- Dawid, A. P., Stone, M., and Zidek, J. V. (1973), “Marginalization Paradoxes in Bayesian and Structural Inference,” Journal of the Royal Statistical Society, Series B, 35, 189–233. [1346] MR0365805
- Del Moral, P., Doucet, A., and Jasra, A. (2006), “Sequential Monte Carlo Samplers,” Journal of the Royal Statistical Society, Series B, 68, 411–436. [1357] MR2278333
- Dempster, A. P. (1966), “New Methods for Reasoning Towards Posterior Distributions Based on Sample Data,” The Annals of Mathematical Statistics, 37, 355–374. [1346] MR0187357
- Dempster, A. P. (1968), “A Generalization of Bayesian Inference” (with discussion), Journal of the Royal Statistical Society, Series B, 30, 205–247. [1346] MR0238428
- Dempster, A. P. (2008), “The Dempster-Shafer Calculus for Statisticians,” International Journal of Approximate Reasoning, 48, 365–377. [1346, 1348] MR2419025
- Douc, R., and Moulines, E. (2008), “Limit Theorems for Weighted Samples With Applications to Sequential Monte Carlo Methods,” The Annals of Statistics, 36, 2344–2376. [1357] MR2458190
- Doucet, A., De Freitas, N., and Gordon, N. (2001), Sequential Monte Carlo Methods in Practice, New York: Springer. [1357] MR1847784
- E, L., Hannig, J., and Iyer, H. K. (2008), “Fiducial Intervals for Variance Components in an Unbalanced Two-Component Normal Mixed Linear Model,” Journal of the American Statistical Association, 103, 854–865. [1347] MR2524335
- E, L., Hannig, J., and Iyer, H. K. (2009), “Applications of Generalized Fiducial Inference,” Ph.D. Dissertation, Colorado State University, Fort Collins, CO. [1348, 1357]
- Edlefsen, P. T., Liu, C., and Dempster, A. P. (2009), “Estimating Limits From Poisson Counting Data Using Dempster–Shafer Analysis,” The Annals of Applied Statistics, 3, 764–790. [1346] MR2750681
- Efron, B. (1998), “R.A. Fisher in the 21st Century,” Statistical Science, 13, 95–122. [1356] MR1767915
- Fan, J., and Lv, J. (2008), “Sure Independence Screening for Ultrahigh Dimensional Feature Space,” Journal of the Royal Statistical Society, Series B, 70, 849–911. [1355] MR2530322
- Fisher, R. A. (1922), “On the Mathematical Foundations of Theoretical Statistics,” Philosophical Transactions of the Royal Society of London, Series A, 222, 309–368. [1346]
- Fisher, R. A. (1925), “Theory of Statistical Estimation,” Proceedings of the Cambridge Philosophical Society, 22, 700–725. [1346]
- Fisher, R. A. (1930), “Inverse Probability,” Proceedings of the Cambridge Philosophical Society, xxvi, 528–535. [1346]
- Fisher, R. A. (1933), “The Concepts of Inverse Probability and Fiducial Probability Referring to Unknown Parameters,” Proceedings of the Royal Society of London, Series A, 139, 343–348. [1346]
- Fisher, R. A. (1935), “The Fiducial Argument in Statistical Inference,” The Annals of Eugenics, VI, 91–98. [1346]
- Fraser, A. M., Fraser, D. A. S., and Staicu, A.-M. (2009), “The Second Order Ancillary: A Differential View With Continuity,” Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 16, 1208–1223. [1347] MR2759176
- Fraser, D., and Naderi, A. (2008), “Exponential Models: Approximations for Probabilities,” Biometrika, 94, 1–9. [1347]
- Fraser, D., Reid, N., and Wong, A. (2005), “What a Model With Data Says About Theta,” International Journal of Statistical Science, 3, 163–178. [1347]
- Fraser, D. A. S. (1961a), “On Fiducial Inference,” The Annals of Mathematical Statistics, 32, 661–676. [1346] MR0130755
- Fraser, D. A. S. (1961b), “The Fiducial Method and Invariance,” Biometrika, 48, 261–280. [1346] MR0133910
- Fraser, D. A. S. (1966), “Structural Probability and a Generalization,” Biometrika, 53, 1–9. [1346] MR0196840
- Fraser, D. A. S. (1968), The Structure of Inference, New York-London-Sydney: Wiley. [1346] MR0235643
- Fraser, D. A. S. (2004), “Ancillaries and Conditional Inference,” Statistical Science, 19, 333–369. [1347] MR2140544
- Fraser, D. A. S. (2011), “Is Bayes Posterior Just Quick and Dirty Confidence?” Statistical Science, 26, 299–316. [1347] MR2918001
- Fraser, D. A. S., Reid, N., Marras, E., and Yi, G. Y. (2010), “Default Priors for Bayesian and Frequentist Inference,” Journal of the Royal Statistical Society, Series B, 72. [1347, 1350] MR2758239
- 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, Fort Collins, CO. [1347]
- Hannig, J. (2009), “On Generalized Fiducial Inference,” Statistica Sinica, 19, 491–544. [1347, 1348, 1349, 1352, 1356, 1358] MR2514173
- Hannig, J. (2013), “Generalized Fiducial Inference via Discretization,” Statistica Sinica, 23, 489–514. [1347, 1348, 1349, 1350, 1357] MR3086644
- Hannig, J. (2014), Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle” by D. G. Mayo, Statistical Science, 29, 254–258. [1358] MR3264539
- Hannig, J., E, L., Abdel-Karim, A., and Iyer, H. K. (2006a), “Simultaneous Fiducial Generalized Confidence Intervals for Ratios of Means of Lognormal Distributions,” Austrian Journal of Statistics, 35, 261–269. [1347, 1358]
- Hannig, J., Iyer, H. K., and Patterson, P. (2006b), “Fiducial Generalized Confidence Intervals,” Journal of American Statistical Association, 101, 254–269. [1347] MR2268043
- Hannig, J., Iyer, H. K., and Wang, J. C.-M. (2007), “Fiducial Approach to Uncertainty Assessment: Accounting for Error Due to Instrument Resolution,” Metrologia, 44, 476–483. [1347]
- Hannig, J., Lai, R. C. S., and Lee, T. C. M. (2014), “Computational Issues of Generalized Fiducial Inference,” Computational Statistics and Data Analysis, 71, 849–858. [1358] MR3132011
- Hannig, J., and Lee, T. C. M. (2009), “Generalized Fiducial Inference for Wavelet Regression,” Biometrika, 96, 847–860. [1347, 1354, 1355, 1358] MR2767274
- Hannig, J., Wang, C. M., and Iyer, H. K. (2003), “Uncertainty Calculation for the Ratio of Dependent Measurements,” Metrologia, 4, 177–186. [1347]
- Hannig, J., and Xie, M. (2012), “A Note on Dempster-Shafer Recombinations of Confidence Distributions,” Electrical Journal of Statistics, 6, 1943–1966. [1347, 1356] MR2988470
- Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999), “Bayesian Model Averaging: A Tutorial” (with discussion), Statistical Science, 14, 382–417; corrected version available at http://www.stat.washington.edu/www/research/online/hoetingl999.pdf. [1354] MR1765176
- Iyer, H. K., and Patterson, P. (2002), “A Recipe for Constructing Generalized Pivotal Quantities and Generalized Confidence Intervals,” Technical Report 10, Department of Statistics, Colorado State University, Fort Collins CO. [1350]
- Iyer, H. K., Wang, C. M. J., and Mathew, T. (2004), “Models and Confidence Intervals for True Values in Interlaboratory Trials,” Journal of the American Statistical Association, 99, 1060–1071. [1347] MR2109495
- Jeffreys, H. (1940), “Note on the Behrens-Fisher Formula,” The Annals of Eugenics, 10, 48–51. [1346] MR0002080
- Lai, R. C. S., Hannig, J., and Lee, T. C. M. (2015), “Generalized Fiducial Inference for Ultra-High Dimensional Regression,” Journal of American Statistical Association, 110, 760–772. [1347, 1354, 1355] MR3367262
- Lawrence, E., Liu, C., Vander Wiel, S., and Zhang, J. (2009), “A New Method for Multinomial Inference Using Dempster-Shafer Theory.” [1356]
- Lee, T. C. M. (2002), “Tree-Based Wavelet Regression for Correlated Data using the Minimum Description Length Principle,” Australian and New Zealand Journal of Statistics, 44, 23–39. [1355] MR1894978
- Lindley, D. V. (1958), “Fiducial Distributions and Bayes’ Theorem,” Journal of the Royal Statistical Society, Series B, 20, 102–107. [1346] MR0095550
- Liu, Y., and Hannig, J. (2016), “Generalized Fiducial Inference for Binary Logistic Item Response Models,” Psychometrica, 81, 290–324. [1347] MR3505368
- Majumder, P. A., and Hannig, J. (2015), “Higher Order Asymptotics for Generalized Fiducial Inference,” unpublished manuscript. [1347, 1353]
- Martin, R., and Liu, C. (2013), “Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference,” Journal of the American Statistical Association, 108, 301–313. [1346] MR3174621
- Martin, R., and Liu, C. (2015a), “Conditional Inferential Models: Combining Information for Prior-Free Probabilistic Inference,” Journal of the Royal Statistical Society, Series B, 77, 195–217. [1346, 1353] MR3299405
- Martin, R., and Liu, C. (2015b), Inferential Models: Reasoning with Uncertainty (Vol. 145), Boca Raton, FL: CRC Press. [1346]
- Martin, R., and Liu, C. (2015c), “Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters,” Journal of the American Statistical Association, 110, 1621–1631. [1346] MR3449059
- Martin, R., and Walker, S. G. (2014), “Asymptotically Minimax Empirical Bayes Estimation of a Sparse Normal Mean Vector,” Electronic Journal of Statistics, 8, 2188–2206. [1353] MR3273623
- Martin, R., Zhang, J., and Liu, C. (2010), “Dempster-Shafer Theory and Statistical Inference With Weak Beliefs,” Statistical Science, 25, 72–87. [1346] MR2741815
- McNally, R. J., Iyer, H. K., and Mathew, T. (2003), “Tests for Individual and Population Bioequivalence Based on Generalized p-Values,” Statistics in Medicine, 22, 31–53. [1347]
- Patterson, P., Hannig, J., and Iyer, H. K. (2004), “Fiducial Generalized Confidence Intervals for Proportionof Conformance,” Technical Report 2004/11, Colorado State University, Fort Collins, CO. [1347]
- Salome, D. (1998), “Statistical Inference via Fiducial Methods,” Ph.D. dissertation, University of Groningen, Groningen, The Netherlands. [1346]
- Schweder, T., and Hjort, N. L. (2002), “Confidence and Likelihood,” Scandinavian Journal of Statistics, 29, 309–332. [1346, 1356] MR1909788
- Singh, K., Xie, M., and Strawderman, W. E. (2005), “Combining Information From Independent Sources Through Confidence Distributions,” The Annals of Statistics, 33, 159–183. [1346] MR2157800
- Sonderegger, D., and Hannig, J. (2014), “Fiducial Theory for Free-Knot Splines,” in Contemporary Developments in Statistical Theory, a Festschrift in honor of Professor Hira L. Koul, eds. S. Lahiri, A. Schick, A. SenGupta, T. N. Sriram, New York: Springer, pp. 155–189. [1347, 1352] MR3149911
- Stevens, W. L. (1950), “Fiducial Limits of the Parameter of a Discontinuous Distribution,” Biometrika, 37, 117–129. [1346] MR0035955
- Taraldsen, G., and Lindqvist, B. H. (2013), “Fiducial Theory and Optimal Inference,” The Annals of Statistics, 41, 323–341. [1347] MR3059420
- Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288. [1358] MR1379242
- Tsui, K.-W., and Weerahandi, S. (1989), “Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters,” Journal of the American Statistical Association, 84, 602–607. [1347] MR1010352
- Tsui, K.-W., and Weerahandi, S. (1991), “Corrections: “Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters,” Journal of the American Statistical Association, 86, 256 (Journal of the American Statistical Association, 84, 1989, 602–607). [1347] MR1010352
- Tukey, J. W. (1957), “Some Examples With Fiducial Relevance,” The Annals of Mathematical Statistics, 28, 687–695. [1346] MR0113257
- Veronese, P., and Melilli, E. (2015), “Fiducial and Confidence Distributions for Real Exponential Families,” Scandinavian Journal of Statistics, 42, 471–484. [1347] MR3345116
- Wandler, D. V., and Hannig, J. (2011), “Fiducial Inference on the Maximum Mean of a Multivariate Normal Distribution,” Journal of Multivariate Analysis, 102, 87–104. [1347] MR2729422
- Wandler, D. V., and Hannig, J. (2012a), “A Fiducial Approach to Multiple Comparisons,” Journal of Statistical Planning and Inference, 142, 878–895. [1347] MR2863876
- Wandler, D. V., and Hannig, J. (2012b), “Generalized Fiducial Confidence Intervals for Extremes,” Extremes, 15, 67–87. [1347, 1358] MR2891310
- Wang, J. C.-M., Hannig, J., and Iyer, H. K. (2012a), “Fiducial Prediction Intervals,” Journal of Statistical Planning and Inference, 142, 1980–1990. [1347, 1353] MR2903406
- Wang, J. C.-M., Hannig, J., and Iyer, H. K. (2012b), “Pivotal Methods in the Propagation of Distributions,” Metrologia, 49, 382–389. [1347]
- Wang, J. C.-M., and Iyer, H. K. (2005), “Propagation of Uncertainties in Measurements Using Generalized Inference,” Metrologia, 42, 145–153. [1347]
- Wang, J. 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. [1347]
- Wang, J. C.-M., and Iyer, H. K. (2006b), “Uncertainty Analysis for Vector Measurands Using Fiducial Inference,” Metrologia, 43, 486–494. [1347]
- Wang, Y. H. (2000), “Fiducial Intervals: What Are They?” The American Statistician, 54, 105–111. [1347] MR1803120
- Weerahandi, S. (1993), “Generalized Confidence Intervals,” Journal of the American Statistical Association, 88, 899–905. [1347] MR1242940
- Weerahandi, S. (1994), Correction: “Generalized Confidence Intervals,” Journal of the American Statistical Association, 89, 726 (Journal of the American Statistical Association, 88, 1993, 899–905). [1347] MR1242940
- Weerahandi, S. (1995), Exact Statistical Methods for Data Analysis, Springer Series in Statistics, New York: Springer-Verlag. [1347] MR1316663
- Welch, B. L., and Peers, H. W. (1963), “On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods,” Journal of the Royal Statistical Society, Series B, 25, 318–329. [1353] MR0173309
- Wilkinson, G. N. (1977), “On Resolving the Controversy in Statistical Inference,” Journal of the Royal Statistical Society, Series B, 39, 119–171. [1346] MR0652326
- Xie, M., Liu, R. Y., Damaraju, C. V., and Olson, W. H. (2013), “Incorporating External Information in Analyses of Clinical Trials With Binary Outcomes,” The Annals of Applied Statistics, 7, 342–368. [1347] MR3086422
- Xie, M., and Singh, K. (2013), “Confidence Distribution, the Frequentist Distribution Estimator of a Parameter: A Review,” International Statistical Review, 81, 3–39. [1346, 1348] MR3047496
- Xie, M., Singh, K., and Strawderman, W. E. (2011), “Confidence Distributions and a Unified Framework for Meta-Analysis,” Journal of the American Statistical Association, 106, 320–333. [1346] MR2816724
- Xu, X., and Li, G. (2006), “Fiducial Inference in the Pivotal Family of Distributions,” Science in China, Series A, 49, 410–432. [1347] MR2223670
- Yang, R., and Berger, J. O. (1997), “A Catalogue of Noninformative Priors,” Technical Report ISDS 97–42, Duke University, Durham, NC. [1350]
- Zhang, C., and Huang, J. (2008), “The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression,” The Annals of Statistics, 36, 1567–1594. [1355] MR2435448
- Zhang, J., and Liu, C. (2011), “Dempster-Shafer Inference With Weak Beliefs,” Statistica Sinica, 21, 475–494. [1346] MR2829843

This list reflects references listed in the original paper as accurately as possible with no attempt to correct error.

This review first appeared at “Generalized fiducial inference: a review and new results. (English summary)” (Mathematical Reviews) with the exception of the clarifying “{MR3149921}” and is used with permission from the American Mathematical Society.

Categories: fiducial inference, reviews

You must be logged in to post a comment.