Priors via imaginary training samples of sufficient statistics for objective Bayesian hypothesis testing. (English) Zbl 1437.62091
Summary: The expected-posterior prior (EPP) and the power-expected-posterior (PEP) prior are based on random imaginary observations and offer several advantages in objective Bayesian hypothesis testing. The use of sufficient statistics, when these exist, as a way to redefine the EPP and PEP prior is investigated. In this way the dimensionality of the problem can be reduced, by generating samples of sufficient statistics instead of generating full sets of imaginary data. On the theoretical side it is proved that the new EPP and PEP definitions based on imaginary training samples of sufficient statistics are equivalent with the standard definitions based on individual training samples. This equivalence provides a strong justification and generalization of the definition of both EPP and PEP prior, since from the individual samples or from the sufficient samples the criteria coincide. This avoids potential inconsistencies or paradoxes when only sufficient statistics are available. The applicability of the new definitions in different hypotheses testing problems is explored, including the case of an irregular model. Calculations are simplified; and it is shown that when testing the mean of a normal distribution the EPP and PEP prior can be expressed as a beta mixture of normal priors. The paper concludes with a discussion about the interpretation and the benefits of the proposed approach.
MSC:
| 62F03 | Parametric hypothesis testing |
| 62F15 | Bayesian inference |
| 62B05 | Sufficient statistics and fields |
Keywords:
Bayesian hypothesis testing; expected-posterior priors; imaginary training samples; objective priors; power-expected-posterior priors; sufficient statisticsReferences:
| [1] | Bartlett, M., Comment on D. V. Lindley’s statistical paradox, Biometrika, 44, 533-534 (1957) · Zbl 0081.13801 · doi:10.1093/biomet/44.3-4.533 |
| [2] | Bayarri, M.; Garcia-Donato, G., Generalization of Jeffreys divergence-based priors for Bayesian hypothesis testing, J. R. Stat. Soc. B, 70, 981-1003 (2008) · Zbl 1411.62042 · doi:10.1111/j.1467-9868.2008.00667.x |
| [3] | Berger, J.; Bernardo, J.; Sun, D., The formal definition of reference priors, Ann. Stat., 37, 905-938 (2009) · Zbl 1162.62013 · doi:10.1214/07-AOS587 |
| [4] | Berger, J.; Pericchi, L., The intrinsic Bayes factor for model selection and prediction, J. Am. Stat. Assoc., 91, 109-122 (1996) · Zbl 0870.62021 · doi:10.1080/01621459.1996.10476668 |
| [5] | Berger, J.; Pericchi, L., Accurate and stable Bayesian model selection: the median intrinsic Bayes factor, Sankhyā Indian J. Stat. Spec. Issue Bayesian Anal., 60, 1-18 (1998) · Zbl 1081.62517 |
| [6] | Berger, J.; Pericchi, L., Training samples in objective model selection, Ann. Stat., 32, 841-869 (2004) · Zbl 1092.62034 · doi:10.1214/009053604000000229 |
| [7] | Bernardo, J.; Rueda, R., Bayesian hypothesis testing: a reference approach, Int. Stat. Rev., 70, 351-372 (2002) · Zbl 1211.62011 · doi:10.1111/j.1751-5823.2002.tb00175.x |
| [8] | Consonni, G.; Fouskakis, D.; Liseo, B.; Ntzoufras, I., Prior distributions for objective Bayesian analysis, Bayesian Anal., 13, 627-679 (2018) · Zbl 1407.62073 · doi:10.1214/18-BA1103 |
| [9] | Consonni, G.; Veronese, P., Compatibility of prior specifications across linear models, Stat. Sci., 23, 332-353 (2008) · Zbl 1329.62331 · doi:10.1214/08-STS258 |
| [10] | Fouskakis, D.; Ntzoufras, I., Power-conditional-expected priors. Using g-priors with random imaginary data for variable selection, J. Comput. Gr. Stat., 25, 647-664 (2015) · doi:10.1080/10618600.2015.1036996 |
| [11] | Fouskakis, D.; Ntzoufras, I., Limiting behavior of the Jeffreys power-expected-posterior Bayes factor in Gaussian linear models, Braz. J. Probab. Stat., 30, 299-320 (2016) · Zbl 1381.62230 · doi:10.1214/15-BJPS281 |
| [12] | Fouskakis, D.; Ntzoufras, I., Information consistency of the Jeffreys power-expected-posterior prior in Gaussian linear models, Metron, 75, 371-380 (2017) · Zbl 1392.62071 · doi:10.1007/s40300-017-0110-6 |
| [13] | Fouskakis, D.; Ntzoufras, I.; Draper, D., Power-expected-posterior priors for variable selection in Gaussian linear models, Bayesian Anal., 10, 75-107 (2015) · Zbl 1335.62045 · doi:10.1214/14-BA887 |
| [14] | Fouskakis, D.; Ntzoufras, I.; Perrakis, K., Power-expected-posterior priors for generalized linear models, Bayesian Anal., 13, 721-748 (2018) · Zbl 1407.62275 · doi:10.1214/17-BA1066 |
| [15] | Good, I.: Probability and the Weighting of Evidence. Haffner, New York (2004) |
| [16] | Griffin, J.; Brown, P., Hierarchical shrinkage priors for regression models, Bayesian Anal., 12, 135-159 (2017) · Zbl 1384.62225 · doi:10.1214/15-BA990 |
| [17] | Ibrahim, J.; Chen, M., Power prior distributions for regression models, Stat. Sci., 15, 46-60 (2000) · doi:10.1214/ss/1009212673 |
| [18] | Johnson, VE; Rossell, D., On the use of non-local prior densities in Bayesian hypothesis tests, J. R. Stat. Soc. Ser. B, 72, 143-170 (2010) · Zbl 1411.62019 · doi:10.1111/j.1467-9868.2009.00730.x |
| [19] | Kass, R.; Wasserman, L., A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion, J. Am. Stat. Assoc., 90, 928-934 (1995) · Zbl 0851.62020 · doi:10.1080/01621459.1995.10476592 |
| [20] | Lourenzutti, R., Duarte, D., Azevedo, M.: The Beta Truncated Pareto Distribution. Technical Report, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil (2014) |
| [21] | Pérez, J.; Berger, J., Expected-posterior prior distributions for model selection, Biometrika, 89, 491-511 (2002) · Zbl 1036.62026 · doi:10.1093/biomet/89.3.491 |
| [22] | Simpson, D.; Rue, H.; Riebler, A.; Martins, T.; Sørbye, S., Penalising model component complexity: a principled, practical approach to constructing priors, Stat. Sci., 32, 1-28 (2017) · Zbl 1442.62060 · doi:10.1214/16-STS576 |
| [23] | Spiegelhalter, D., Abrams, K., Myles, J.: Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Statistics in Practice. Wiley, Chichester (2004) |
| [24] | Spiegelhalter, D.; Smith, A., Bayes factors for linear and log-linear models with vague prior information, J. R. Stat. Soc. Ser. B, 44, 377-387 (1982) · Zbl 0502.62032 |
| [25] | Zellner, A.; Goel, P. (ed.); Zellner, A. (ed.), On assessing prior distributions and Bayesian regression analysis using g-prior distributions, 233-243 (1986), Amsterdam · Zbl 0608.00012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
