A Mass-Shifting Phenomenon of Truncated Multivariate Normal Priors

References

• Azzalini, A., and Valle, A. D. (1996), “The Multivariate Skew-Normal Distribution,” Biometrika, 83, 715–726. DOI: 10.1093/biomet/83.4.715.
   Web of Science ®Google Scholar
• Bai, R., Moran, G. E., Antonelli, J. L., Chen, Y., and Boland, M. R. (2020), “Spike-and-Slab Group LASSOs for Grouped Regression and Sparse Generalized Additive Models,” Journal of the American Statistical Association, 117, 184–197. DOI: 10.1080/01621459.2020.1765784.
   Web of Science ®Google Scholar
• Bhattacharya, A., Dunson, D. B., Pati, D., and Pillai, N. S. (2016), “Sub-Optimality of Some Continuous Shrinkage Priors,” Stochastic Processes and their Applications, 126, 3828–3842. DOI: 10.1016/j.spa.2016.08.007.
   Web of Science ®Google Scholar
• Bickel, P., and Lindner, M. (2012), “Approximating the Inverse of Banded Matrices by Banded Matrices with Applications to Probability and Statistics,” Theory of Probability & Its Applications, 56, 1–20.
   Web of Science ®Google Scholar
• Bickel, P. J., and Levina, E. (2008), “Regularized Estimation of Large Covariance Matrices,” The Annals of Statistics, 36, 199–227. DOI: 10.1214/009053607000000758.
   Web of Science ®Google Scholar
• Bornkamp, B., and Ickstadt, K. (2009), “Bayesian Nonparametric Estimation of Continuous Monotone Functions with Applications to Dose–Response Analysis,” Biometrics, 65, 198–205. DOI: 10.1111/j.1541-0420.2008.01060.x.
   PubMed Web of Science ®Google Scholar
• Botev, Z. I. (2017), “The Normal Law under Linear Restrictions: Simulation and Estimation via Minimax Tilting,” Journal of the Royal Statistical Society, Series B, 79, 125–148. DOI: 10.1111/rssb.12162.
   Google Scholar
• Boucheron, S., Lugosi, G., and Massart, P. (2013), Concentration Inequalities: A Nonasymptotic Theory of Independence, Oxford: Oxford University Press.
   Google Scholar
• Brezger, A., and Steiner, W. J. (2008), “Monotonic Regression based on Bayesian P–Splines: An Application to Estimating Price Response Functions from Store-Level Scanner Data,” Journal of Business & Economic Statistics, 26, 90–104.
   Web of Science ®Google Scholar
• Cai, B., and Dunson, D. B. (2007), “Bayesian Multivariate Isotonic Regression Splines: Applications to Carcinogenicity Studies,” Journal of the American Statistical Association, 102, 1158–1171. DOI: 10.1198/016214506000000942.
   Web of Science ®Google Scholar
• Candes, E. J., Romberg, J. K., and Tao, T. (2006), “Stable Signal Recovery from Incomplete and Inaccurate Measurements,” Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59, 1207–1223. DOI: 10.1002/cpa.20124.
   Google Scholar
• Cartinhour, J. (1990), “One-Dimensional Marginal Density Functions of a Truncated Multivariate Normal Density Function,” Communications in Statistics-Theory and Methods, 19, 197–203. DOI: 10.1080/03610929008830197.
   Web of Science ®Google Scholar
• Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010), “The Horseshoe Estimator for Sparse Signals,” Biometrika, 97, 465–480. DOI: 10.1093/biomet/asq017.
   Web of Science ®Google Scholar
• Castillo, I., and van der Vaart, A. (2012), “Needles and Straw in a Haystack: Posterior Concentration for Possibly Sparse Sequences,” The Annals of Statistics, 40, 2069–2101. DOI: 10.1214/12-AOS1029.
   Web of Science ®Google Scholar
• Castillo, I., Schmidt-Hieber, J., and van der Vaart, A. (2015), “Bayesian Linear Regression with Sparse Priors,” The Annals of Statistics, 43, 1986–2018. DOI: 10.1214/15-AOS1334.
   Web of Science ®Google Scholar
• Chakraborty, A., Bhattacharya, A., and Mallick, B. K. (2020), “Bayesian Sparse Multiple Regression for Simultaneous Rank Reduction and Variable Selection,” Biometrika, 107, 205–221. DOI: 10.1093/biomet/asz056.
   PubMed Web of Science ®Google Scholar
• Chakraborty, M., and Ghosal, S. (2021), “Convergence Rates for Bayesian Estimation and Testing in Monotone Regression,” Electronic Journal of Statistics, 15, 3478–3503. DOI: 10.1214/21-EJS1861.
   Web of Science ®Google Scholar
• Chatterjee, S., Guntuboyina, A., and Sen, B. (2015), “On Risk Bounds in Isotonic and other Shape Restricted Regression Problems,” The Annals of Statistics, 43, 1774–1800. DOI: 10.1214/15-AOS1324.
   Web of Science ®Google Scholar
• Chipman, H. A., George, E. I., and McCulloch, R. E. (2010), “BART: Bayesian Additive Regression Trees,” The Annals of Applied Statistics, 4, 266–298. DOI: 10.1214/09-AOAS285.
   Web of Science ®Google Scholar
• Curtis, S. M., and Ghosh, S. K. (2011), “A Variable Selection Approach to Monotonic Regression with Bernstein Polynomials,” Journal of Applied Statistics, 38, 961–976. DOI: 10.1080/02664761003692423.
   Web of Science ®Google Scholar
• Duan, L. L., Young, A. L., Nishimura, A., and Dunson, D. B. (2020), “Bayesian Constraint Relaxation,” Biometrika, 107, 191–204. DOI: 10.1093/biomet/asz069.
   PubMed Web of Science ®Google Scholar
• Dunson, D. B. (2005), “Bayesian Semiparametric Isotonic Regression for Count Data,” Journal of the American Statistical Association, 100, 618–627. DOI: 10.1198/016214504000001457.
   Web of Science ®Google Scholar
• Gao, C., Han, F., and Zhang, C.-H. (2020), “On Estimation of Isotonic Piecewise Constant Signals,” The Annals of Statistics, 48, 629–654. DOI: 10.1214/18-AOS1792.
   Web of Science ®Google Scholar
• Ghosal, S., Ghosh, J. K., and van der Vaart, A. (2000), “Convergence Rates of Posterior Distributions,” The Annals of Statistics, 28, 500–531. DOI: 10.1214/aos/1016218228.
   Web of Science ®Google Scholar
• Lenk, P. J., and Choi, T. (2017), “Bayesian Analysis of Shape-Restricted Functions using Gaussian Process Priors,” Statistica Sinica, 27, 43–69.
   Web of Science ®Google Scholar
• Lin, L., and Dunson, D.B. (2014), “Bayesian Monotone Regression using Gaussian Process Projection,” Biometrika, 101, 303–317. DOI: 10.1093/biomet/ast063.
   Web of Science ®Google Scholar
• Maatouk, H., and Bay, X. (2017), “Gaussian Process Emulators for Computer Experiments with Inequality Constraints,” Mathematical Geosciences, 49, 557–582. DOI: 10.1007/s11004-017-9673-2.
   Web of Science ®Google Scholar
• Meyer, M. C., Hackstadt, A. J., and Hoeting, J. A. (2011), “Bayesian Estimation and Inference for Generalised Partial Linear Models using Shape-Restricted Splines,” Journal of Nonparametric Statistics, 23, 867–884. DOI: 10.1080/10485252.2011.597852.
   Web of Science ®Google Scholar
• Narisetty, N. N., and He, X. (2014), “Bayesian Variable Selection with Shrinking and Diffusing Priors,” The Annals of Statistics, 42, 789–817. DOI: 10.1214/14-AOS1207.
   Web of Science ®Google Scholar
• Neelon, B., and Dunson, D. B. (2004), “Bayesian Isotonic Regression and Trend Analysis,” Biometrics, 60, 398–406. DOI: 10.1111/j.0006-341X.2004.00184.x.
   PubMed Web of Science ®Google Scholar
• Pati, D., Bhattacharya, A., Pillai, N. S., and Dunson, D. (2014), “Posterior Contraction in Sparse Bayesian Factor Models for Massive Covariance Matrices,” The Annals of Statistics, 42, 1102–1130. DOI: 10.1214/14-AOS1215.
   Web of Science ®Google Scholar
• Polson, N. G., and Scott, J. G. (2010), “Shrink Globally, Act Locally: Sparse Bayesian Regularization and Prediction,” Bayesian Statistics, 9, 501–538.
   Google Scholar
• Polson, N. G., and Scott, J. G. (2012), “On the Half-Cauchy Prior for a Global Scale Parameter,” Bayesian Analysis, 7, 887–902.
   Web of Science ®Google Scholar
• Ray, P., Pati, D., and Bhattacharya, A. (2020), “Efficient Bayesian Shape-Restricted Function Estimation with Constrained Gaussian Process Priors,” Statistics and Computing, 30, 839–853. DOI: 10.1007/s11222-020-09922-0.
   Web of Science ®Google Scholar
• Sen, D., Patra, S., and Dunson, D. (2018), “Constrained Inference through Posterior Projections,” arXiv preprint arXiv:1812.05741.
   Google Scholar
• Shively, T. S., Walker, S. G., and Damien, P. (2011), “Nonparametric Function Estimation Subject to Monotonicity, Convexity and other Shape Constraints,” Journal of Econometrics, 161, 166–181. DOI: 10.1016/j.jeconom.2010.12.001.
   Web of Science ®Google Scholar
• Song, Q., and Liang, F. (2017), “Nearly Optimal Bayesian Shrinkage for High Dimensional Regression,” arXiv preprint arXiv:1712.08964.
   Google Scholar
• Sun, D., and Berger, J. O. (1998), “Reference Priors with Partial Information,” Biometrika, 85, 55–71. DOI: 10.1093/biomet/85.1.55.
   Web of Science ®Google Scholar
• Talagrand, M. (1995), “Concentration of Measure and Isoperimetric Inequalities in Product Spaces,” Publications Mathématiques de l’Institut des Hautes Etudes Scientifiques, 81, 73–205. DOI: 10.1007/BF02699376.
   Google Scholar
• Van der Vaart, A. W. (2000), Asymptotic Statistics (Vol. 3), Cambridge: Cambridge University Press.
   Google Scholar
• Wei, R., and Ghosal, S. (2020), “Contraction Properties of Shrinkage Priors in Logistic Regression,” Journal of Statistical Planning and Inference, 207, 215–229. DOI: 10.1016/j.jspi.2019.12.004.
   Web of Science ®Google Scholar
• Yoo, W. W., and Ghosal, S. (2016), “Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression,” The Annals of Statistics, 44, 1069–1102. DOI: 10.1214/15-AOS1398.
   Web of Science ®Google Scholar