Statistical Science

Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”

Jan Hannig

Full-text: Open access

Abstract

In this discussion we demonstrate that fiducial distributions provide a natural example of an inference paradigm that does not obey Strong Likelihood Principle while still satisfying the Weak Conditionality Principle.

Article information

Source
Statist. Sci., Volume 29, Number 2 (2014), 254-258.

Dates
First available in Project Euclid: 18 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ss/1408368578

Digital Object Identifier
doi:10.1214/14-STS474

Mathematical Reviews number (MathSciNet)
MR3264539

Zentralblatt MATH identifier
1332.62023

Keywords
Generalized fiducial inference strong likelihood principle violation weak conditionality principle

Citation

Hannig, Jan. Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle”. Statist. Sci. 29 (2014), no. 2, 254--258. doi:10.1214/14-STS474. https://projecteuclid.org/euclid.ss/1408368578


Export citation

References

  • Barnard, G. A. (1995). Pivotal models and the fiducial argument. Internat. Statist. Rev. 63 309–323.
  • Bayarri, M. J., Berger, J. O., Forte, A. and García-Donato, G. (2012). Criteria for Bayesian model choice with application to variable selection. Ann. Statist. 40 1550–1577.
  • Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors. In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 35–60. Oxford Univ. Press, New York.
  • Berger, J. O., Bernardo, J. M. and Sun, D. (2009). The formal definition of reference priors. Ann. Statist. 37 905–938.
  • Berger, J. O., Bernardo, J. M. and Sun, D. (2012). Objective priors for discrete parameter spaces. J. Amer. Statist. Assoc. 107 636–648.
  • Berger, J. O. and Sun, D. (2008). Objective priors for the bivariate normal model. Ann. Statist. 36 963–982.
  • Chiang, A. K. L. (2001). A simple general method for constructing confidence intervals for functions of variance components. Technometrics 43 356–367.
  • Cisewski, J. and Hannig, J. (2012). Generalized fiducial inference for normal linear mixed models. Ann. Statist. 40 2102–2127.
  • Cox, D. R. (1958). Some problems connected with statistical inference. Ann. Math. Statist. 29 357–372.
  • Dawid, A. P. and Stone, M. (1982). The functional-model basis of fiducial inference. Ann. Statist. 10 1054–1074.
  • Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 35 189–233.
  • 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. R. Stat. Soc. Ser. B Stat. Methodol. 30 205–247.
  • Dempster, A. P. (2008). The Dempster–Shafer calculus for statisticians. Internat. J. Approx. Reason. 48 365–377.
  • Edlefsen, P. T., Liu, C. and Dempster, A. P. (2009). Estimating limits from Poisson counting data using Dempster–Shafer analysis. Ann. Appl. Stat. 3 764–790.
  • Efron, B. (1998). R. A. Fisher in the 21st century (invited paper presented at the 1996 R. A. Fisher Lecture). Statist. Sci. 13 95–122.
  • Fisher, R. A. (1930). Inverse probability. Math. Proc. Cambridge Philos. Soc. XXVI 528–535.
  • Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to unknown parameters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 139 343–348.
  • Fisher, R. A. (1935). The fiducial argument in statistical inference. Ann. Eugenics VI 91–98.
  • Fraser, D. A. S. (1961a). On fiducial inference. Ann. Math. Statist. 32 661–676.
  • Fraser, D. A. S. (1961b). The fiducial method and invariance. Biometrika 48 261–280.
  • Fraser, D. A. S. (1966). Structural probability and a generalization. Biometrika 53 1–9.
  • Fraser, D. A. S. (1968). The Structure of Inference. Wiley, New York.
  • Fraser, D. A. S. (2004). Ancillaries and conditional inference. Statist. Sci. 19 333–369.
  • Fraser, D. A. S. (2011). Is Bayes posterior just quick and dirty confidence? Statist. Sci. 26 299–316.
  • Fraser, A. M., Fraser, D. A. S. and Staicu, A.-M. (2010). Second order ancillary: A differential view from continuity. Bernoulli 16 1208–1223.
  • Fraser, D. A. S. and Naderi, A. (2008). Exponential models: Approximations for probabilities. Biometrika 94 1–9.
  • Fraser, D., Reid, N. and Wong, A. (2005). What a model with data says about theta. Internat. J. Statist. Sci. 3 163–178.
  • Fraser, D. A. S., Reid, N., Marras, E. and Yi, G. Y. (2010). Default priors for Bayesian and frequentist inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 631–654.
  • Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica 19 491–544.
  • Hannig, J. (2013). Generalized fiducial inference via discretization. Statist. Sinica 23 489–514.
  • Hannig, J., Iyer, H. and Patterson, P. (2006). Fiducial generalized confidence intervals. J. Amer. Statist. Assoc. 101 254–269.
  • Hannig, J., Lai, R. C. S. and Lee, T. C. M. (2014). Computational issues of generalized fiducial inference. Comput. Statist. Data Anal. 71 849–858.
  • Hannig, J. and Lee, T. C. M. (2009). Generalized fiducial inference for wavelet regression. Biometrika 96 847–860.
  • Hannig, J. and Xie, M.-g. (2012). A note on Dempster–Shafer recombination of confidence distributions. Electron. J. Stat. 6 1943–1966.
  • 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.
  • Lai, R. C. S., Hannig, J. and Lee, T. C. M. (2013). Generalized fiducial inference for ultra high dimensional regression. Available at arXiv:1304.7847.
  • Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. R. Stat. Soc. Ser. B Stat. Methodol. 20 102–107.
  • Martin, R. and Liu, C. (2013a). Conditional inferential models: Combining information for prior-free probabilistic inference. Preprint.
  • Martin, R. and Liu, C. (2013b). Inferential models: A framework for prior-free posterior probabilistic inference. J. Amer. Statist. Assoc. 108 301–313.
  • Martin, R. and Liu, C. (2013c). Marginal inferential models: prior-free probabilistic inference on interest parameters. Preprint.
  • Martin, R. and Liu, C. (2013d). On a ’plausible’ interpretation of p-values. Preprint.
  • Martin, R., Zhang, J. and Liu, C. (2010). Dempster–Shafer theory and statistical inference with weak beliefs. Statist. Sci. 25 72–87.
  • Patterson, P., Hannig, J. and Iyer, H. K. (2004). Fiducial generalized confidence intervals for proportion of conformance. Technical Report 2004/11, Colorado State Univ., Fort Collins, CO.
  • Salome, D. (1998). Staristical inference via fiducial methods. Ph.D. thesis, Univ. Groningen.
  • Schweder, T. and Hjort, N. L. (2002). Confidence and likelihood. Scand. J. Stat. 29 309–332. Large structured models in applied sciences; challenges for statistics (Grimstad, 2000).
  • Singh, K., Xie, M. and Strawderman, W. E. (2005). Combining information from independent sources through confidence distributions. Ann. Statist. 33 159–183.
  • Sonderegger, D. and Hannig, J. (2014). Fiducial theory for free-knot splines. In Contemporaly Developments in Statistical Theory, a Festschrift in Honor of Professor Hira L. Koul (T. N. Sriraus, ed.) 155–189. Springer, Berlin.
  • Stevens, W. L. (1950). Fiducial limits of the parameter of a discontinuous distribution. Biometrika 37 117–129.
  • Taraldsen, G. and Lindqvist, B. H. (2013). Fiducial theory and optimal inference. Ann. Statist. 41 323–341.
  • Tsui, K.-W. and Weerahandi, S. (1989). Generalized $p$-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84 602–607.
  • Tsui, K.-W. and Weerahandi, S. (1991). Corrections: “Generalized $p$-values in significance testing of hypotheses in the presence of nuisance parameters”. [J. Amer. Statist. Assoc. 84 (1989) 602–607. MR1010352]. J. Amer. Statist. Assoc. 86 256.
  • Wang, C. M., Hannig, J. and Iyer, H. K. (2012). Fiducial prediction intervals. J. Statist. Plann. Inference 142 1980–1990.
  • Weerahandi, S. (1993). Generalized confidence intervals. J. Amer. Statist. Assoc. 88 899–905.
  • Weerahandi, S. (1994). Correction: “Generalized confidence intervals”. [J. Amer. Statist. Assoc. 88 (1993) 899–905. MR1242940]. J. Amer. Statist. Assoc. 89 726.
  • Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. Springer Series in Statistics. Springer, New York.
  • Wilkinson, G. N. (1977). On resolving the controversy in statistical inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 39 119–171.
  • Xie, M.-g. and Singh, K. (2013). Confidence distribution, the frequentist distribution estimator of a parameter: A review. Internat. Statist. Rev. 81 3–39.
  • Xie, M., Singh, K. and Strawderman, W. E. (2011). Confidence distributions and a unifying framework for meta-analysis. J. Amer. Statist. Assoc. 106 320–333.
  • Xie, M., Liu, R. Y., Damaraju, C. V. and Olson, W. H. (2013). Incorporating external information in analyses of clinical trials with binary outcomes. Ann. Appl. Stat. 7 342–368.
  • Zhang, J. and Liu, C. (2011). Dempster–Shafer inference with weak beliefs. Statist. Sinica 21 475–494.

See also

  • Main article: On the Birnbaum Argument for the Strong Likelihood Principle.