High-dimensional properties for empirical priors in linear regression with unknown error variance. (English) Zbl 07840076
Summary: We study full Bayesian procedures for high-dimensional linear regression. We adopt data-dependent empirical priors introduced in Martin et al. (Bernoulli 23(3):1822-1847, 2017). In their paper, these priors have nice posterior contraction properties and are easy to compute. Our paper extend their theoretical results to the case of unknown error variance. Under proper sparsity assumption, we achieve model selection consistency, posterior contraction rates as well as Bernstein von-Mises theorem by analyzing multivariate t-distribution.
MSC:
| 62-XX | Statistics |
Keywords:
Bernstein von-Mises theorem; model selection consistency; multivariate t-distribution; posterior contraction rateReferences:
| [1] | Arias-Castro, E.; Lounici, K., Estimation and variable selection with exponential weights, Electron J Stat, 8, 1, 328-354 (2014) · Zbl 1294.62164 · doi:10.1214/14-EJS883 |
| [2] | Bondell, H.; Reich, B., Consistent high-dimensional Bayesian variable selection via penalized credible regions, J Am Stat Assoc, 107, 1601-1624 (2012) · Zbl 1258.62026 · doi:10.1080/01621459.2012.716344 |
| [3] | Cao, X.; Khare, K.; Ghosh, M., High-dimensional posterior consistency for hierarchical non-local priors in regression, Bayesian Anal, 15, 1, 241-262 (2020) · Zbl 1437.62253 · doi:10.1214/19-BA1154 |
| [4] | Castillo, I.; Schmidt-Hieber, J.; van der vaart, A., Bayesian linear regression with sparse priors, Ann Stat, 43, 5, 1986-2018 (2015) · Zbl 1486.62197 · doi:10.1214/15-AOS1334 |
| [5] | de Geer, SAV; Buhlmann, P., On the conditions used to prove oracle results for the lasso, Electron J Stat, 3, 1360-1392 (2009) · Zbl 1327.62425 |
| [6] | Grünwald, P.; van Ommen, T., Inconsistency of Bayesian inference for misspecified linear models, and a proposal for repairing it, Bayesian Anal, 12, 1069-1103 (2017) · Zbl 1384.62088 · doi:10.1214/17-BA1085 |
| [7] | Holmes, CC; Walker, SG, Assigning a value to a power likelihood in a general Bayesian model, Biometrika, 104, 2, 497-503 (2017) · Zbl 1506.62264 |
| [8] | Hong, L.; Kuffner, TA; Martin, R., On overfitting and post-selection uncertainty assessments, Biometrika, 105, 1, 221-224 (2018) · Zbl 07072407 · doi:10.1093/biomet/asx083 |
| [9] | Ishwaran, H.; Rao, J., Spike and slab variable selection: frequentist and Bayesian strategies, Ann Stat, 33, 730-773 (2005) · Zbl 1068.62079 · doi:10.1214/009053604000001147 |
| [10] | Johnson, VE; Rossell, D., Bayesian model selection in high-dimensional settings, J Am Stat Assoc, 107, 649-660 (2012) · Zbl 1261.62024 · doi:10.1080/01621459.2012.682536 |
| [11] | Martin, R.; Ning, B., Empirical priors and coverage of posterior credible sets in a sparse normal mean model, Indian J Stat, 82-A, 477-498 (2020) · Zbl 1451.62017 |
| [12] | Martin, R.; Tang, Y., Empirical priors for prediction in sparse high-dimensional linear regression, J Mach Learn Res, 21, 1-30 (2020) · Zbl 1519.62014 |
| [13] | Martin, R.; Walker, S., Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector, Electron J Stat, 8, 2, 2188-2206 (2014) · Zbl 1302.62015 · doi:10.1214/14-EJS949 |
| [14] | Martin, R.; Walker, S., Data-dependent priors and their posterior concentration rates, Electron J Stat, 13, 3049-3081 (2019) · Zbl 1429.62148 · doi:10.1214/19-EJS1600 |
| [15] | Martin, R.; Mess, R.; Walker, S., Empirical bayes posterior concentration in sparse high-dimensional linear models, Bernoulli, 23, 3, 1822-1847 (2017) · Zbl 1450.62085 · doi:10.3150/15-BEJ797 |
| [16] | Narisetty, N.; He, X., Bayesian variable selection with shrinking and diffusing priors, Ann Stat, 42, 789-817 (2014) · Zbl 1302.62158 · doi:10.1214/14-AOS1207 |
| [17] | Shin, M.; Bhattacharya, A.; Johnson, VE, Functional horseshoe priors for subspace shrinkage, J Am Stat Assoc, 115, 532, 1784-1797 (2019) · Zbl 1452.62523 · doi:10.1080/01621459.2019.1654875 |
| [18] | Syring, M.; Martin, R., Calibrating general posterior credible regions, Biometrika, 106, 479-486 (2019) · Zbl 1454.62105 · doi:10.1093/biomet/asy054 |
| [19] | Wainwright, MJ, Sharp thresholds for high-dimensional and noisy sparsity recovery using l1 -constrained quadratic programming (lasso), IEEE Trans, 55, 2183-2202 (2009) · Zbl 1367.62220 |
| [20] | Zou, H., The adaptive lasso and its oracle properties, J Am Stat Assoc, 101, 1418-1429 (2006) · Zbl 1171.62326 · doi:10.1198/016214506000000735 |
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.
