Bayesian Analysis
- Bayesian Anal.
- Volume 9, Number 4 (2014), 939-962.
Equivalence between the Posterior Distribution of the Likelihood Ratio and a p-value in an Invariant Frame
Isabelle Smith and André Ferrari
Full-text: Open access
Abstract
The Posterior distribution of the Likelihood Ratio (PLR) is proposed by Dempster in 1973 for significance testing in the simple vs. composite hypothesis case. In this hypothesis test case, classical frequentist and Bayesian hypothesis tests are irreconcilable, as emphasized by Lindley’s paradox, Berger & Selke in 1987 and many others. However, Dempster shows that the PLR (with inner threshold 1) is equal to the frequentist p-value in the simple Gaussian case. In 1997, Aitkin extends this result by adding a nuisance parameter and showing its asymptotic validity under more general distributions. Here we extend the reconciliation between the PLR and a frequentist p-value for a finite sample, through a framework analogous to the Stein’s theorem frame in which a credible (Bayesian) domain is equal to a confidence (frequentist) domain.
Article information
Source
Bayesian Anal., Volume 9, Number 4 (2014), 939-962.
Dates
First available in Project Euclid: 21 November 2014
Permanent link to this document
https://projecteuclid.org/euclid.ba/1416579186
Digital Object Identifier
doi:10.1214/14-BA877
Mathematical Reviews number (MathSciNet)
MR3293963
Zentralblatt MATH identifier
1327.62165
Keywords
hypothesis testing PLR p-value likelihood ratio frequentist and Bayesian reconciliation Lindley’s paradox invariance
Citation
Smith, Isabelle; Ferrari, André. Equivalence between the Posterior Distribution of the Likelihood Ratio and a p-value in an Invariant Frame. Bayesian Anal. 9 (2014), no. 4, 939--962. doi:10.1214/14-BA877. https://projecteuclid.org/euclid.ba/1416579186
References
- Aitkin, M. (1997). “The calibration of p-values, posterior Bayes factors and the AIC from the posterior distribution of the likelihood.” Statistics and Computing, 7: 253–261.
- — (2010). Statistical inference: an integrated Bayesian / likelihood approach. Chapman and Hall.Mathematical Reviews (MathSciNet): MR2677456
- Aitkin, M., Boys, R. J., and Chadwick, T. (2005). “Bayesian point null hypothesis testing via the posterior likelihood ratio.” Statistics and Computing, 25(3): 217–230.Mathematical Reviews (MathSciNet): MR2147554
Digital Object Identifier: doi:10.1007/s11222-005-1310-0 - Aitkin, M., Liu, C. C., and Chadwick, T. (2009). “Bayesian model comparison and model averaging for small-area estimation.” Annals of Applied Statistics, 3(1): 199–221.Mathematical Reviews (MathSciNet): MR2668705
Digital Object Identifier: doi:10.1214/08-AOAS205
Project Euclid: euclid.aoas/1239888368 - Baskurt, Z. and Evans, M. (2013). “Hypothesis assessment and inequalities for Bayes factors and relative belief ratios.” Bayesian Analysis, 8,3: 569–590.Mathematical Reviews (MathSciNet): MR3102226
Digital Object Identifier: doi:10.1214/13-BA824
Project Euclid: euclid.ba/1378729920 - Berger, J. and Sellke, T. (1987). “Testing a point null hypothesis: the irreconcilability of P values and evidence (with discussion).” Journal of the American Statistical Association, 82: 112–139.Mathematical Reviews (MathSciNet): MR883340
- Berger, J. O. (1985). Statistical decision theory and Bayesian analysis. Springer-Verlag, 2nd edition.Mathematical Reviews (MathSciNet): MR804611
- Berger, J. O., Brown, L., and Wolpert, R. (1994). “A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing.” Annals of Statistics, 22(4): 1787–1807.Mathematical Reviews (MathSciNet): MR1329168
Digital Object Identifier: doi:10.1214/aos/1176325757
Project Euclid: euclid.aos/1176325757 - Berger, J. O. and Delampady, M. (1987). “Testing precise hypotheses (with discussion).” Statistical Science, 2(3): 317–335.Mathematical Reviews (MathSciNet): MR920141
Digital Object Identifier: doi:10.1214/ss/1177013238
Project Euclid: euclid.ss/1177013238 - Bernardo, J. (2011). Bayesian Statistics 9, chapter Integrated objective Bayesian estimation and hypothesis testing. Oxford University Press.Mathematical Reviews (MathSciNet): MR3204001
Digital Object Identifier: doi:10.1093/acprof:oso/9780199694587.003.0001 - Birnbaum, A. (1962). “On the foundation of statistical inference (with discussion).” Journal of the American Statistical Association, 57(298): 269–326.Mathematical Reviews (MathSciNet): MR138176
Digital Object Identifier: doi:10.1080/01621459.1962.10480660 - Borges, W. and Stern, J. (2007). “The rules of logic composition for the Bayesian epistemic e-values.” Logic journal of the IGPL, 15(5–6): 401–420.
- Casella, G. and Berger, R. L. (1987). “Reconciling Bayesian and frequentist evidence in the one-sided testing problem.” Journal of the American Statistical Association, 82(397): 106–111.Mathematical Reviews (MathSciNet): MR883339
Digital Object Identifier: doi:10.1080/01621459.1987.10478396 - Chang, T. and Villegas, C. (1986). “On a theorem of Stein relating Bayesian and classical inferences in group models.” The Canadian Journal of Statistics, 14(4): 289–296.
- Darmois, G. (1935). “Sur les lois de probabilité à estimation exhaustive.” Compte-Rendu de l’Académie des Sciences de Paris, 200(1265–1266).
- Dempster, A. P. (1973). “The direct use of likelihood for significance testing.” In Proceedings of Conference on Foundational Questions in Statistical Inference, 335–354. Aaarhus, Denmark.Mathematical Reviews (MathSciNet): MR408052
- — (1997). “Commentary on the paper by Murray Aitkin, and on discussion by Mervyn Stone.” Statistics and Computing, 7(4): 265–269.
- Eaton, M. (1989). Group invariance applications in statistics. Regional Conference Series in Probability and Statistics.
- — (2007). Multivariate statistics. Institute of Mathematical Statistics.
- Eaton, M. and Sudderth, W. (1999). “Consistency and strong inconsistency of group-invariant predictive inferences.” Bernoulli, 5(5): 833–854.Mathematical Reviews (MathSciNet): MR1715441
Digital Object Identifier: doi:10.2307/3318446
Project Euclid: euclid.bj/1171290401 - — (2002). “Group invariant inference and right Haar measure.” Journal of Statistical Planning and Inference, 103(1–2): 87–99.Mathematical Reviews (MathSciNet): MR1896985
Digital Object Identifier: doi:10.1016/S0378-3758(01)00199-9 - Evans, M. (1997). “Bayesian inference procedures derived via the concept of relative surprise.” Communications in Statistics, 26: 1125–1143.Mathematical Reviews (MathSciNet): MR1450226
Digital Object Identifier: doi:10.1080/03610929708831972 - Evans, M. and Jang, G. (2010). “Invariant p-values for model checking.” Annals of Statistics, 38(1): 512–525.Mathematical Reviews (MathSciNet): MR2589329
Digital Object Identifier: doi:10.1214/09-AOS727
Project Euclid: euclid.aos/1262271622 - Fisher, R. A. (1973, 1st ed.: 1956). Statistical methods and scientific inference. Oliver and Boyd, 3rd edition.Mathematical Reviews (MathSciNet): MR346955
- Fraser, D. A. S. (1961). “The fiducial method and invariance.” Biometrika, 48(3–4): 261–280.Mathematical Reviews (MathSciNet): MR133910
Digital Object Identifier: doi:10.1093/biomet/48.3-4.261 - Goutis, C. and Casella, G. (1997). “Relationships between post-data accuracy measures.” Annals of the Institute of Statistical Mathematics, 49(4): 711–726.
- Hwang, J., Casella, G., Robert, C., Wells, M., and Farrell, R. (1992). “Estimation of accuracy in testing.” Annals of Statistics, 20(1): 490–509.Mathematical Reviews (MathSciNet): MR1150356
Digital Object Identifier: doi:10.1214/aos/1176348534
Project Euclid: euclid.aos/1176348534 - Jeffreys, H. (1961). Theory of probability. Oxford University Press, 3rd edition.Mathematical Reviews (MathSciNet): MR187257
- Lehmann, E. L. and Romano, J. P. (2005). Testing statistical hypotheses. Springer, 3rd edition.Mathematical Reviews (MathSciNet): MR2135927
- Lindley, D. (1957). “A statistical paradox.” Biometrika, 44(1–2): 187–192.
- Meng, X.-L. (1994). “Posterior predictive p-values.” Annals of Statistics, 22(3): 1142–1160.Mathematical Reviews (MathSciNet): MR1311969
Digital Object Identifier: doi:10.1214/aos/1176325622
Project Euclid: euclid.aos/1176325622 - Nachbin, L. (1965). The Haar integral. Van Nostrand.Mathematical Reviews (MathSciNet): MR175995
- Newton, M. and Raftery, A. (1994). “Approximate Bayesian inference with the weighted likelihood bootstrap.” Journal of the Royal Statistical Society Series B, 56(1): 3–48.Mathematical Reviews (MathSciNet): MR1257793
- Neyman, J. (1977). “Frequentist probability and frequentist statistics.” Synthese, 36: 97–131.
- Neyman, J. and Pearson, E. (1933). “On the problem of the most efficient tests of statistical hypotheses.” Philosophical Transactions of the Royal Society of London, Series A, 231: 289–337.
- Oh, H. and DasGupta, A. (1999). “Comparison of the p-value and posterior probability.” Journal of Statistical Planning and Inference, 76(1–2): 93–107.Mathematical Reviews (MathSciNet): MR1673342
Digital Object Identifier: doi:10.1016/S0378-3758(98)00131-1 - O’Hagan, A. (1995). “Fractional Bayes factors for model comparison.” Journal of the Royal Statistical Society, 57(1): 99–138.Mathematical Reviews (MathSciNet): MR1325379
- Pereira, C. and Stern, J. (1999). “Evidence and credibility: full Bayesian significance test for precise hypotheses.” Entropy, 1: 104–115.
- Robert, C. P. (2007). The Bayesian choice. Springer, 2nd edition.Mathematical Reviews (MathSciNet): MR2723361
- Robins, J., van der Vaart, A., and Ventura, V. (2000). “Asymptotic distribution of p-values in composite null models.” Journal of the American Statistical Association, 95(452): 1143–1156.Mathematical Reviews (MathSciNet): MR1804240
Digital Object Identifier: doi:10.1080/01621459.2000.10474214 - Royall, R. (1997). Statistical evidence: a likelihood paradigm. Chapman and Hall / CRC Press.Mathematical Reviews (MathSciNet): MR1629481
- Sellke, T., Bayarri, M. J., and Berger, J. O. (2001). “Calibration of p-values for testing precise null hypotheses.” American Statistician, 55(1): 62—71.Mathematical Reviews (MathSciNet): MR1818723
Digital Object Identifier: doi:10.1198/000313001300339950 - Smith, I. (2010). “Détection d’une source faible : modèles et méthodes statistiques. Application à la détection d’exoplanètes par imagerie directe.” Ph.D. thesis, Université de Nice Sophia-Antipolis.
- Smith, I. and Ferrari, A. (2010). “The posterior distribution of the likelihood ratio as a measure of evidence.” In International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 391–398.Mathematical Reviews (MathSciNet): MR2857066
- Stein, C. (1965). “Approximation of improper prior measures by prior probability measures.” In Bernoulli, Bayes, Laplace Festschrift, 217–240. Springer-Verlag.Mathematical Reviews (MathSciNet): MR199937
- Tsao, C. A. (2006). “A note on Lindley’s paradox.” Test, 15(1): 125–139.
- Villegas, C. (1981). “Inner statistical inference II.” Annals of Statistics, 9(4): 768–776.Mathematical Reviews (MathSciNet): MR619280
Digital Object Identifier: doi:10.1214/aos/1176345517
Project Euclid: euclid.aos/1176345517 - Zidek, J. (1969). “A representation of Bayesian invariant procedures in terms of Haar measure.” Annals of the Institute of Statistical Mathematics, 21(1): 291–308.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Posterior Predictive $p$-Values
Meng, Xiao-Li, The Annals of Statistics, 1994 - The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain
Dickey, James M. and Lientz, B. P., The Annals of Mathematical Statistics, 1970 - A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing
Berger, James O., Brown, Lawrence D., and Wolpert, Robert L., The Annals of Statistics, 1994
- Posterior Predictive $p$-Values
Meng, Xiao-Li, The Annals of Statistics, 1994 - The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain
Dickey, James M. and Lientz, B. P., The Annals of Mathematical Statistics, 1970 - A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing
Berger, James O., Brown, Lawrence D., and Wolpert, Robert L., The Annals of Statistics, 1994 - General frequentist properties of the posterior profile distribution
Cheng, Guang and Kosorok, Michael R., The Annals of Statistics, 2008 - Unified frequentist and Bayesian testing of a precise hypothesis
Berger, J. O., Boukai, B., and Wang, Y., Statistical Science, 1997 - A case for robust Bayesian priors with applications to clinical trials
Cook, John D., Fúquene, Jairo A., and Pericchi, Luis R., Bayesian Analysis, 2009 - Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes
Bickel, David R., Electronic Journal of Statistics, 2012 - On Optimum Tests of Composite Hypotheses with One Constraint
Lehmann, E. L., The Annals of Mathematical Statistics, 1947 - Domains of Optimality of Tests in Simple Random Sampling
Hildebrand, David K., The Annals of Mathematical Statistics, 1969 - Self-consistent confidence sets and tests of composite hypotheses applicable to restricted parameters
Bickel, David R. and Patriota, Alexandre G., Bernoulli, 2019