Abstract
We investigate the asymptotic behavior of posterior distributions in nonparametric regression problems when the distribution of noise structure of the regression model is assumed to be non-Gaussian but symmetric such as the Laplace distribution. Given prior distributions for the unknown regression function and the scale parameter of noise distribution, we show that the posterior distribution concentrates around the true values of parameters. Following the approach by Choi and Schervish (Journal of Multivariate Analysis, 98, 1969–1987, 2007) and extending their results, we prove consistency of the posterior distribution of the parameters for the nonparametric regression when errors are symmetric non-Gaussian with suitable assumptions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amewou-Atisso M., Ghosal S., Ghosh J.K., Ramamoorthi R.V. (2003). Posterior consistency for semiparametric regression problems. Bernoulli 9:291–312
Anderson T.W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proceedings of the American Mathematical Society 6:170–176
Barron A.R. (1989). Uniformly powerful goodness of fit tests. The Annals of Statistics 17:107–124
Barron A.R., Schervish M.J., Wasserman L. (1999). The consistency of posterior distributions in nonparametric problems. The Annals of Statistics 27:536–561
Chernoff H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on a sum of observations. The Annals of Mathematical Statistics 23:493–507
Choi, T. (2005). Posterior consistency in nonparametric regression problems under Gaussian process priors. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh.
Choi T. (2007). Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors. Journal of Statistical Planning and Inference 137:2975–2983
Choi T., Schervish M.J. (2007). On posterior consistency in nonparametric regression problems. Journal of Multivariate Analysis 98: 1969–1987
Choudhuri N., Ghosal S., Roy A. (2004). Bayesian estimation of the spectral density of a time series. Journal of the American Statistical Association 99:1050–1059
Diaconis P., Freedman D. (1986). On inconsistent Bayes estimates. The Annals of Statistics 14:68–87
Freedman D. (1963). On the asymptotic distribution of Bayes estimates in the discrete case I. The Annals of Mathematical Statistics 34:1386–1403
Ge Y., Jiang W. (2006). On consistency of Bayesian inference with mixtures of logistic regression. Neural Computation 18:224–243
Ghosal S., Roy A. (2006). Posterior consistency of Gaussian process prior for nonparametric binary regression. The Annals of Statistics 34:2413–2429
Ghosal, S., van der Vaart, A.W. Convergence rates of posterior distributions for noniid observations. The Annals of Statistics, 35, 192–223.
Ghosh J.K., Ramamoorthi R.V. (2003). Bayesian nonparametrics. New York, Springer
Huang T.M. (2004). Convergence rates for posterior distributions and adaptive estimation. The Annals of Statistics 32:1556–1593
Kleijn B.J.K., van der Vaart A.W. (2006). Misspecification in infinite-dimensional Bayesian statistics. The Annals of Statistics 34:837–877
Kotz, S., Kozubowski, T. J., Podgorski, K. (2001). Laplace distributions and generalizations: a revisit with applications to communications, economics, engineering and finance. Birkhäuser: Springer.
Neal, R. M. (1996). Bayesian learning for neural networks. In: Lecture notes in Statistics (vol. 118). New York: Springer.
Neal, R. M. (1997). Monte Carlo implementation of Gaussian process models for Bayesian regression and classification. Technical Report, 9702, Department of Statistics, University of Toronto.
Paciorek, C. J. (2003). Nonstationary Gaussian processes for regression and spatial modelling. Ph.D. Thesis. Carnegie Mellon University, Pittsburgh.
Schwartz L. (1965). On Bayes procedures. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 4:10–26
Shen X., Wasserman L. (2001). Rates of convergence of posterior distributions. The Annals of Statistics 29:687–714
Stone C.J. (1977). Consistent nonparametric regression. The Annals of Statistics 5:595–620
Tokdar S., Ghosh J.K. (2007). Posterior consistency of Gaussian process priors in density estimation. Journal of Statistical Planning and Inference Inference 137:34–42
Walker S. (2004). New approach to Bayesian consistency. The Annals of Statistics 32:2028–2043
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Choi, T. Asymptotic properties of posterior distributions in nonparametric regression with non-Gaussian errors. Ann Inst Stat Math 61, 835–859 (2009). https://doi.org/10.1007/s10463-008-0168-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-008-0168-2