Effect of Model Space Priors on Statistical Inference with Model Uncertainty
Volume 1, Issue 2 (2023), pp. 149–158
Pub. online: 16 November 2022
Type: Statistical Methodology
Open Access
Open Access
Accepted
18 October 2022
18 October 2022
Published
16 November 2022
16 November 2022
Abstract
Bayesian model averaging (BMA) provides a coherent way to account for model uncertainty in statistical inference tasks. BMA requires specification of model space priors and parameter space priors. In this article we focus on comparing different model space priors in the presence of model uncertainty. We consider eight reference model space priors used in the literature and three adaptive parameter priors recommended by Porwal and Raftery [37]. We assess the performance of these combinations of prior specifications for variable selection in linear regression models for the statistical tasks of parameter estimation, interval estimation, inference, point and interval prediction. We carry out an extensive simulation study based on 14 real datasets representing a range of situations encountered in practice. We found that beta-binomial model space priors specified in terms of the prior probability of model size performed best on average across various statistical tasks and datasets, outperforming priors that were uniform across models. Recently proposed complexity priors performed relatively poorly.
Supplementary material
Supplementary MaterialThe supplementary material contains detailed summary results for each metric and dataset used in the study. It also contains a summary of data-generating models for each of the datasets.
References
Bartlett, M. S. (1957). A Comment on D. V. Lindley’s Statistical Paradox. Biometrika 44 533–534. https://doi.org/10.1093/biomet/52.3-4.507. MR0207142
Castillo, I., Schmidt-Hieber, J. and Van der Vaart, A. (2015). Bayesian linear regression with sparse priors. The Annals of Statistics 43(5) 1986–2018. https://doi.org/10.1214/15-AOS1334. MR3375874
Celeux, G., El Anbari, M., Marin, J. q. M. and Robert, C. P. (2012). Regularization in Regression: Comparing Bayesian and Frequentist Methods in a Poorly Informative Situation. Bayesian Analysis 7 477–502. https://doi.org/10.1214/12-BA716. MR2934959
Clyde, M. and George, E. I. (2000). Flexible empirical Bayes estimation for wavelets. Journal of the Royal Statistical Society: Series B, Statistical Methodology 62(4) 681–698. https://doi.org/10.1111/1467-9868.00257. MR1796285
Clyde, M. and George, E. I. (2004). Model uncertainty. Statistical Science 19(1) 81–94. https://doi.org/10.1214/088342304000000035. MR2082148
Dellaportas, P., Forster, J. J. and Ntzoufras, I. (2012). Joint specification of model space and parameter space prior distributions. Statistical Science 27(2) 232–246. https://doi.org/10.1214/11-STS369. MR2963994
Eicher, T. S., Papageorgiou, C. and Raftery, A. E. (2011). Default priors and predictive performance in Bayesian model averaging, with application to growth determinants. Journal of Applied Econometrics 26(1) 30–55. https://doi.org/10.1002/jae.1112. MR2759908
Fan, J. and Lv, J. (2008). Sure independence screening for ultrahigh dimensional feature space. Journal of the Royal Statistical Society: Series B — Statistical Methodology 70(5) 849–911. https://doi.org/10.1111/j.1467-9868.2008.00674.x. MR2530322
Fernández, C., Ley, E. and Steel, M. F. J. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics 100(2) 381–427. https://doi.org/10.1016/S0304-4076(00)00076-2. MR1820410
Filzmoser, P. and Varmuza, K. (2017). chemometrics: Multivariate Statistical Analysis in Chemometrics. R package version 1.4.2. https://CRAN.R-project.org/package=chemometrics.
Forte, A., Garcia-Donato, G. and Steel, M. F. J. (2018). Methods and tools for Bayesian variable selection and model averaging in normal linear regression. International Statistical Review 86(2) 237–258. https://doi.org/10.1111/insr.12249. MR3852410
Foster, D. P. and George, E. I. (1994). The risk inflation criterion for multiple regression. Annals of Statistics 22(4) 1947–1975. https://doi.org/10.1214/aos/1176325766. MR1329177
George, E. (1999). Discussion of “Model averaging and model search strategies” by M. Clyde. In Bayesian Statistics 6–Proceedings of the Sixth Valencia International Meeting. MR1723497
George, E. I. (2010). Dilution priors: Compensating for model space redundancy. In Borrowing Strength: Theory Powering Applications–A Festschrift for Lawrence D. Brown 158–165 Institute of Mathematical Statistics. MR2798517
George, E. I. and Foster, D. P. (2000). Calibration and empirical Bayes variable selection. Biometrika 87(4) 731–747. https://doi.org/10.1093/biomet/87.4.731. MR1813972
Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102(477) 359–378. https://doi.org/10.1198/016214506000001437. MR2345548
Hansen, M. H. and Yu, B. (2003). Minimum description length model selection criteria for generalized linear models. Lecture Notes-Monograph Series 40 145–163. https://doi.org/10.1214/lnms/1215091140. MR2004337
Hoeting, J. A., Raftery, A. E. and Madigan, D. (2002). Bayesian variable and transformation selection in linear regression. Journal of Computational and Graphical Statistics 11(3) 485–507. https://doi.org/10.1198/106186002501. MR1938444
Hoeting, J. A., Madigan, D., Raftery, A. E. and Volinsky, C. T. (1999). Bayesian model averaging: a tutorial. Statistical Science 14 382–417. https://doi.org/10.1214/ss/1009212519. MR1765176
Ishwaran, H., Rao, J. S. and Kogalur, U. B. (2013). spikeslab: Prediction and variable selection using spike and slab regression. R package version 1.1.5. http://cran.r-project.org/web/packages/spikeslab/. https://doi.org/10.1214/21-ejp733. MR4366222
James, G., Witten, D., Hastie, T. and Tibshirani, R. (2017). ISLR: Data for an Introduction to Statistical Learning with Applications in R. R package version 1.2. https://CRAN.R-project.org/package=ISLR. https://doi.org/10.1007/978-1-0716-1418-1. MR4309209
Kass, R. E. and Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association 90(430) 773–795. https://doi.org/10.1080/01621459.1995.10476572. MR3363402
Leamer, E. E. (1978) Specification Searches: Ad hoc Inference with Nonexperimental Data 53. Wiley. MR0471118
Ley, E. and Steel, M. F. (2009). On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. Journal of applied econometrics 24(4) 651–674. https://doi.org/10.1002/jae.1057. MR2675199
Liang, F., Paulo, R., Molina, G., Clyde, M. A. and Berger, J. O. (2008). Mixtures of g priors for Bayesian variable selection. Journal of the American Statistical Association 103 410–423. https://doi.org/10.1198/016214507000001337. MR2420243
Lumley, T. (2020). leaps: Regression Subset Selection. R package version 3.1. https://CRAN.R-project.org/package=leaps.
Narisetty, N. N. and He, X. (2014). Bayesian variable selection with shrinking and diffusing priors. The Annals of Statistics 42(2) 789–817. https://doi.org/10.1214/14-AOS1207. MR3210987
Newman, D. J., Hettich, S., Blake, C. L. and Merz, C. J. (1998). UCI Repository of machine learning databases. http://www.ics.uci.edu/~mlearn/MLRepository.html.
Raftery, A. E. (1988). Approximate Bayes factors for generalized linear models. Technical Report No. 121, Department of Statistics, University of Washington. https://stat.uw.edu/sites/default/files/files/reports/1988/tr121.pdf.
Raftery, A. E., Madigan, D. and Hoeting, J. A. (1997). Bayesian model averaging for linear regression models. Journal of the American Statistical Association 92(437) 179–191. https://doi.org/10.2307/2291462. MR1436107
Rossell, D. (2021). Concentration of posterior model probabilities and normalized l0 criteria. Bayesian Analysis 1(1) 1–27. https://doi.org/10.1214/21-ba1262. MR4483231
Rossell, D. and Rubio, F. J. (2018). Tractable Bayesian Variable Selection: Beyond Normality. Journal of the American Statistical Association. https://doi.org/10.1080/01621459.2017.1371025. MR3902243
Rossell, D., Abril, O. and Bhattacharya, A. (2021). Approximate Laplace approximations for scalable model selection. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 83(4) 853–879. MR4320004
Scott, J. G. and Berger, J. O. (2010). Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. The Annals of Statistics 2587–2619. https://doi.org/10.1214/10-AOS792. MR2722450
van Zwet, E. (2019). A default prior for regression coefficients. Statistical Methods in Medical Research 28(12) 3799–3807. https://doi.org/10.1177/0962280218817792. MR4003623
Villa, C. and Walker, S. (2015). An objective Bayesian criterion to determine model prior probabilities. Scandinavian Journal of Statistics 42(4) 947–966. https://doi.org/10.1111/sjos.12145. MR3426304
Wasserman, L. (2000). Bayesian model selection and model averaging. Journal of Mathematical Psychology 44(1) 92–107. https://doi.org/10.1006/jmps.1999.1278. MR1770003
Yang, Y., Wainwright, M. J. and Jordan, M. I. (2016). On the computational complexity of high-dimensional Bayesian variable selection. The Annals of Statistics 44(6) 2497–2532. https://doi.org/10.1214/15-AOS1417. MR3576552
Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In Bayesian Inference and Decision Techniques 6. MR0881437
