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International Encyclopedia of Statistical Science pp 106–107Cite as

Bayesian Semiparametric Regression

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Linear or generalized linear models assume that the (conditional) mean μ = E(y | x), of the response y, given the covariate vector x, is linked to a linear predictor μ by

$$\mu = h(\eta ),\quad \eta = \mathbf{x}^{\prime}\beta .$$

Here, h is a known response function and βis an unknown vector of regression parameters. More generally, other characteristics of the response distribution, such as variance or skewness may be related to covariates in similar manner. Another example is the Cox model for survival data, where the hazard rate is assumed to have the form

$$\lambda (t\vert \mathbf{x}) = {\lambda }_{0}(t)\exp (\mathbf{x}^{\prime}\beta )$$

with λ0(t) as an (unspecified) baseline hazard rate. In most practical regression situations, however, we are facing at least one of the following problems.

  1. (a)

    For the continuous covariates in the data set, the assumption of a strictly linear effect on the predictor may not be appropriate.

  2. (b)

    Observations may be spatially correlated.

    ...

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References and Further Reading

  • Brezger A, Lang S (2006) Generalized structured additive regression based on Bayesian P-splines. Comput Stat Data Anal 50: 967–991

    MathSciNet  Google Scholar 

  • Denison DGT, Holmes CC, Mallick BK, Smith AFM (2002) Bayesian methods for nonlinear classification and regression. Wiley, Chichester

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  • Fahrmeir L, Kneib T (2010) Bayesian smoothing and regression of longitudinal, spatial and event history data. Oxford University Press, to appear

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  • Fahrmeir L, Kneib T, Lang S (2004) Penalized structured additive regression for space-time data: a Bayesian perspective. Stat Sinica 14:731–761

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  • Hennerfeind A, Brezger A, Fahrmeir L (2006) Geoadditive survival models. J Am Stat Assoc 101:1065–1075

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  • Kneib T, Fahrmeir L (2006) Structured additive regression for multi-categorical space-time data: a mixed model approach. Biometrics 62:109–118

    MATH  MathSciNet  Google Scholar 

  • Kohn R, Smith M, Chan D (2001) Nonparametric regression using linear combinations of basis functions. Stat Comput 11: 313–322

    MathSciNet  Google Scholar 

  • Rue H, Martino S, Chopin N (2009) Approximate Bayesian inference for latent gaussian models by using integrated nested laplace approximations. J R Stat Soc B 71:1–35

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  • Ruppert D, Wand M, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, Cambridge

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  • Ruppert D, Wand MP, Carroll RJ (2009) Semiparametric regression during 2003–2007. Electron J Stat 3:1193–1256

    MATH  MathSciNet  Google Scholar 

  • Smith M, Kohn R, Yau P (2000) Nonparametric Bayesian bivariate surface estimation. In: Schimek G (ed) Smoothing and regression, Ch 19. Wiley, New York

    Google Scholar 

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Author information

Authors and Affiliations

  1. Ludwig-Maximilians-University, Munich, Germany

    Ludwig Fahrmeir (Professor)

Authors
  1. Ludwig Fahrmeir
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Fahrmeir, L. (2011). Bayesian Semiparametric Regression. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_138

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