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On posterior consistency in nonparametric regression problems

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  • Choi, Taeryon
  • Schervish, Mark J.

Abstract

We provide sufficient conditions to establish posterior consistency in nonparametric regression problems with Gaussian errors when suitable prior distributions are used for the unknown regression function and the noise variance. When the prior under consideration satisfies certain properties, the crucial condition for posterior consistency is to construct tests that separate from the outside of the suitable neighborhoods of the parameter. Under appropriate conditions on the regression function, we show there exist tests, of which the type I error and the type II error probabilities are exponentially small for distinguishing the true parameter from the complements of the suitable neighborhoods of the parameter. These sufficient conditions enable us to establish almost sure consistency based on the appropriate metrics with multi-dimensional covariate values fixed in advance or sampled from a probability distribution. We consider several examples of nonparametric regression problems.

Suggested Citation

  • Choi, Taeryon & Schervish, Mark J., 2007. "On posterior consistency in nonparametric regression problems," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1969-1987, November.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:10:p:1969-1987
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    1. Nidhan Choudhuri & Subhashis Ghosal & Anindya Roy, 2004. "Bayesian Estimation of the Spectral Density of a Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1050-1059, December.
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    2. Barde, Sylvain, 2024. "Bayesian estimation of large-scale simulation models with Gaussian process regression surrogates," Computational Statistics & Data Analysis, Elsevier, vol. 196(C).
    3. Debamita Kundu & Riten Mitra & Jeremy T. Gaskins, 2021. "Bayesian variable selection for multioutcome models through shared shrinkage," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 295-320, March.
    4. Liao, Yuan & Jiang, Wenxin, 2011. "Posterior consistency of nonparametric conditional moment restricted models," MPRA Paper 38700, University Library of Munich, Germany.
    5. G. Yi & J. Q. Shi & T. Choi, 2011. "Penalized Gaussian Process Regression and Classification for High-Dimensional Nonlinear Data," Biometrics, The International Biometric Society, vol. 67(4), pages 1285-1294, December.
    6. Cordoni, Francesco & Dorémus, Nicolas & Moneta, Alessio, 2024. "Identification of vector autoregressive models with nonlinear contemporaneous structure," Journal of Economic Dynamics and Control, Elsevier, vol. 162(C).
    7. Seongil Jo & Taeyoung Roh & Taeryon Choi, 2016. "Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 177-206, March.
    8. Marco, Nicholas & Şentürk, Damla & Jeste, Shafali & DiStefano, Charlotte C. & Dickinson, Abigail & Telesca, Donatello, 2024. "Flexible regularized estimation in high-dimensional mixed membership models," Computational Statistics & Data Analysis, Elsevier, vol. 194(C).
    9. Sylvain Barde, 2022. "Bayesian Estimation of Large-Scale Simulation Models with Gaussian Process Regression Surrogates," Studies in Economics 2203, School of Economics, University of Kent.
    10. Pierpaolo De Blasi & Lancelot F. James & John W. Lau, 2007. "Bayesian Nonparametric Estimation and Consistency of Mixed Multinomial Logit Choice Models," ICER Working Papers - Applied Mathematics Series 15-2007, ICER - International Centre for Economic Research.
    11. Jacobi Liana & Kwok Chun Fung & Ramírez-Hassan Andrés & Nghiem Nhung, 2024. "Posterior Manifolds over Prior Parameter Regions: Beyond Pointwise Sensitivity Assessments for Posterior Statistics from MCMC Inference," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 28(2), pages 403-434, April.
    12. Maitra, Trisha & Bhattacharya, Sourabh, 2015. "On Bayesian asymptotics in stochastic differential equations with random effects," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 148-159.
    13. Kim, Gwangsu & Choi, Taeryon, 2019. "Asymptotic properties of nonparametric estimation and quantile regression in Bayesian structural equation models," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 68-82.
    14. Lian, Heng & Choi, Taeryon & Meng, Jie & Jo, Seongil, 2016. "Posterior convergence for Bayesian functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 27-41.
    15. Ximing Wu, 2021. "Hierarchical Gaussian Process Models for Regression Discontinuity/Kink under Sharp and Fuzzy Designs," Papers 2110.00921, arXiv.org, revised Feb 2022.
    16. Weining Shen & Subhashis Ghosal, 2015. "Adaptive Bayesian Procedures Using Random Series Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1194-1213, December.
    17. Debdeep Pati & David Dunson, 2014. "Bayesian nonparametric regression with varying residual density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 1-31, February.

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