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, JanIyer, HariLai, Randy C. S.Lee, Thomas C. M.
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.
In addition to providing an inspiring exposition of generalized fiducial inference, the authors report these new contributions:
1. A weak-limit definition of a generalized fiducial distribution.
2. Sufficient conditions for a generalized fiducial distribution to have asymptotic frequentist coverage.
3. 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

1. Barnard, G. A. (1995), “Pivotal Models and the Fiducial Argument,” International Statistical Reviews, 63, 309–323. [1346]
2. 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
3. Beaumont, M. A., Zhang, W., and Balding, D. J. (2002), “Approximate Bayesian Computation in Population Genetics,” Genetics, 162, 2025–2035. [1349]
4. Berger, J. O. (1992), “On the Development of Reference Priors” (with discussion), Bayesian Statistics, 4, 35–60. [1347] MR1380269
5. 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
6. 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
7. 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
8. 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
9. Berger, J. O., and Sun, D. (2008), “Objective Priors for the Bivariate Normal Model,” The Annals of Statistics, 36, 963–982. [1347] MR2396821
10. Birnbaum, A. (1961), “On the Foundations of Statistical Inference: Binary Experiments,” The Annals of Mathematical Statistics, 32, 414–435. [1348] MR0126307
11. Birnbaum, A. (1962), “On the Foundations of Statistical Inference,” Journal of the American Statistical Association, 57, 269–326. [1350] MR0138176
12. Blom, G. (1976), “Some Properties of Incomplete U-Statistics,” Biometrika, 63, 573–580. [1358] MR0474582
13. 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
14. 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
15. Chen, J., and Chen, Z. (2008), “Extended Bayesian Information Criteria for Model Selection With Large Model Spaces,” Biometrika, 95, 759–771. [1355] MR2443189
16. Chiang, A. K. L. (2001), “A Simple General Method for Constructing Confidence Intervals for Functions of Variance Components,” Technometrics, 43, 356–367. [1347] MR1943189
17. 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
18. Dawid, A. P., and Stone, M. (1982), “The Functional-Model Basis of Fiducial Inference,” The Annals of Statistics, 10, 1054–1074. [1346] MR0673643
19. 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
20. 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
21. 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
22. Dempster, A. P. (1968), “A Generalization of Bayesian Inference” (with discussion), Journal of the Royal Statistical Society, Series B, 30, 205–247. [1346] MR0238428
23. Dempster, A. P. (2008), “The Dempster-Shafer Calculus for Statisticians,” International Journal of Approximate Reasoning, 48, 365–377. [1346, 1348] MR2419025
24. 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
25. Doucet, A., De Freitas, N., and Gordon, N. (2001), Sequential Monte Carlo Methods in Practice, New York: Springer. [1357] MR1847784
26. 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
27. 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]
28. 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
29. Efron, B. (1998), “R.A. Fisher in the 21st Century,” Statistical Science, 13, 95–122. [1356] MR1767915
30. 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
31. 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]
32. Fisher, R. A. (1925), “Theory of Statistical Estimation,” Proceedings of the Cambridge Philosophical Society, 22, 700–725. [1346]
33. Fisher, R. A. (1930), “Inverse Probability,” Proceedings of the Cambridge Philosophical Society, xxvi, 528–535. [1346]
34. 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]
35. Fisher, R. A. (1935), “The Fiducial Argument in Statistical Inference,” The Annals of Eugenics, VI, 91–98. [1346]
36. 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
37. Fraser, D., and Naderi, A. (2008), “Exponential Models: Approximations for Probabilities,” Biometrika, 94, 1–9. [1347]
38. 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]
39. Fraser, D. A. S. (1961a), “On Fiducial Inference,” The Annals of Mathematical Statistics, 32, 661–676. [1346] MR0130755
40. Fraser, D. A. S. (1961b), “The Fiducial Method and Invariance,” Biometrika, 48, 261–280. [1346] MR0133910
41. Fraser, D. A. S. (1966), “Structural Probability and a Generalization,” Biometrika, 53, 1–9. [1346] MR0196840
42. Fraser, D. A. S. (1968), The Structure of Inference, New York-London-Sydney: Wiley. [1346] MR0235643
43. Fraser, D. A. S. (2004), “Ancillaries and Conditional Inference,” Statistical Science, 19, 333–369. [1347] MR2140544
44. Fraser, D. A. S. (2011), “Is Bayes Posterior Just Quick and Dirty Confidence?” Statistical Science, 26, 299–316. [1347] MR2918001
45. 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
46. 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]
47. Hannig, J. (2009), “On Generalized Fiducial Inference,” Statistica Sinica, 19, 491–544. [1347, 1348, 1349, 1352, 1356, 1358] MR2514173
48. Hannig, J. (2013), “Generalized Fiducial Inference via Discretization,” Statistica Sinica, 23, 489–514. [1347, 1348, 1349, 1350, 1357] MR3086644
49. 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
50. 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]
51. Hannig, J., Iyer, H. K., and Patterson, P. (2006b), “Fiducial Generalized Confidence Intervals,” Journal of American Statistical Association, 101, 254–269. [1347] MR2268043
52. 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]
53. 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
54. Hannig, J., and Lee, T. C. M. (2009), “Generalized Fiducial Inference for Wavelet Regression,” Biometrika, 96, 847–860. [1347, 1354, 1355, 1358] MR2767274
55. Hannig, J., Wang, C. M., and Iyer, H. K. (2003), “Uncertainty Calculation for the Ratio of Dependent Measurements,” Metrologia, 4, 177–186. [1347]
56. 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
57. 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
58. 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]
59. 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
60. Jeffreys, H. (1940), “Note on the Behrens-Fisher Formula,” The Annals of Eugenics, 10, 48–51. [1346] MR0002080
61. 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
62. Lawrence, E., Liu, C., Vander Wiel, S., and Zhang, J. (2009), “A New Method for Multinomial Inference Using Dempster-Shafer Theory.” [1356]
63. 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
64. Lindley, D. V. (1958), “Fiducial Distributions and Bayes’ Theorem,” Journal of the Royal Statistical Society, Series B, 20, 102–107. [1346] MR0095550
65. Liu, Y., and Hannig, J. (2016), “Generalized Fiducial Inference for Binary Logistic Item Response Models,” Psychometrica, 81, 290–324. [1347] MR3505368
66. Majumder, P. A., and Hannig, J. (2015), “Higher Order Asymptotics for Generalized Fiducial Inference,” unpublished manuscript. [1347, 1353]
67. 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
68. 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
69. Martin, R., and Liu, C. (2015b), Inferential Models: Reasoning with Uncertainty (Vol. 145), Boca Raton, FL: CRC Press. [1346]
70. 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
71. 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
72. Martin, R., Zhang, J., and Liu, C. (2010), “Dempster-Shafer Theory and Statistical Inference With Weak Beliefs,” Statistical Science, 25, 72–87. [1346] MR2741815
73. 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]
74. 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]
75. Salome, D. (1998), “Statistical Inference via Fiducial Methods,” Ph.D. dissertation, University of Groningen, Groningen, The Netherlands. [1346]
76. Schweder, T., and Hjort, N. L. (2002), “Confidence and Likelihood,” Scandinavian Journal of Statistics, 29, 309–332. [1346, 1356] MR1909788
77. 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
78. 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
79. Stevens, W. L. (1950), “Fiducial Limits of the Parameter of a Discontinuous Distribution,” Biometrika, 37, 117–129. [1346] MR0035955
80. Taraldsen, G., and Lindqvist, B. H. (2013), “Fiducial Theory and Optimal Inference,” The Annals of Statistics, 41, 323–341. [1347] MR3059420
81. Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288. [1358] MR1379242
82. 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
83. 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
84. Tukey, J. W. (1957), “Some Examples With Fiducial Relevance,” The Annals of Mathematical Statistics, 28, 687–695. [1346] MR0113257
85. Veronese, P., and Melilli, E. (2015), “Fiducial and Confidence Distributions for Real Exponential Families,” Scandinavian Journal of Statistics, 42, 471–484. [1347] MR3345116
86. 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
87. Wandler, D. V., and Hannig, J. (2012a), “A Fiducial Approach to Multiple Comparisons,” Journal of Statistical Planning and Inference, 142, 878–895. [1347] MR2863876
88. Wandler, D. V., and Hannig, J. (2012b), “Generalized Fiducial Confidence Intervals for Extremes,” Extremes, 15, 67–87. [1347, 1358] MR2891310
89. 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
90. Wang, J. C.-M., Hannig, J., and Iyer, H. K. (2012b), “Pivotal Methods in the Propagation of Distributions,” Metrologia, 49, 382–389. [1347]
91. Wang, J. C.-M., and Iyer, H. K. (2005), “Propagation of Uncertainties in Measurements Using Generalized Inference,” Metrologia, 42, 145–153. [1347]
92. 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]
93. Wang, J. C.-M., and Iyer, H. K. (2006b), “Uncertainty Analysis for Vector Measurands Using Fiducial Inference,” Metrologia, 43, 486–494. [1347]
94. Wang, Y. H. (2000), “Fiducial Intervals: What Are They?” The American Statistician, 54, 105–111. [1347] MR1803120
95. Weerahandi, S. (1993), “Generalized Confidence Intervals,” Journal of the American Statistical Association, 88, 899–905. [1347] MR1242940
96. 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
97. Weerahandi, S. (1995), Exact Statistical Methods for Data Analysis, Springer Series in Statistics, New York: Springer-Verlag. [1347] MR1316663
98. 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
99. Wilkinson, G. N. (1977), “On Resolving the Controversy in Statistical Inference,” Journal of the Royal Statistical Society, Series B, 39, 119–171. [1346] MR0652326
100. 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
101. 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
102. 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
103. Xu, X., and Li, G. (2006), “Fiducial Inference in the Pivotal Family of Distributions,” Science in China, Series A, 49, 410–432. [1347] MR2223670
104. Yang, R., and Berger, J. O. (1997), “A Catalogue of Noninformative Priors,” Technical Report ISDS 97–42, Duke University, Durham, NC. [1350]
105. 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
106. 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.