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Bayesian regularisation in structured additive regression: a unifying perspective on shrinkage, smoothing and predictor selection

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Abstract

This paper surveys various shrinkage, smoothing and selection priors from a unifying perspective and shows how to combine them for Bayesian regularisation in the general class of structured additive regression models. As a common feature, all regularisation priors are conditionally Gaussian, given further parameters regularising model complexity. Hyperpriors for these parameters encourage shrinkage, smoothness or selection. It is shown that these regularisation (log-) priors can be interpreted as Bayesian analogues of several well-known frequentist penalty terms. Inference can be carried out with unified and computationally efficient MCMC schemes, estimating regularised regression coefficients and basis function coefficients simultaneously with complexity parameters and measuring uncertainty via corresponding marginal posteriors. For variable and function selection we discuss several variants of spike and slab priors which can also be cast into the framework of conditionally Gaussian priors. The performance of the Bayesian regularisation approaches is demonstrated in a hazard regression model and a high-dimensional geoadditive regression model.

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References

  • Baladandayuthapani, V., Mallick, B.K., Carroll, R.J.: Spatially adaptive Bayesian penalized regression splines (P-splines). J. Comput. Graph. Stat. 14, 378–394 (2005)

    Article  MathSciNet  Google Scholar 

  • Besag, J., York, J., Mollié, A.: Bayesian image restoration with two applications in spatial statistics. Ann. Inst. Stat. Math. 43, 1–59 (1991)

    Article  MATH  Google Scholar 

  • Bigelow, J.L., Dunson, D.B.: Bayesian adaptive regression splines for hierarchical data. Biometrics 63, 724–732 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Bondell, H.D., Reich, B.J.: Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR. Biometrics 64, 115–123 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Cai, B., Dunson, D.: Bayesian covariance selection in generalized linear mixed models. Biometrics 62, 446–457 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Casella, G., Moreno, E.: Objective Bayesian variable selection. J. Am. Stat. Assoc. 101, 157–167 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, Z., Dunson, D.: Random effects selection in linear mixed models. Biometrics 59, 762–769 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Chib, S., Jeliazkov, I.: Inference in semiparametric dynamic models for binary longitudinal data. J. Am. Stat. Assoc. 101, 685–700 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Eilers, P.H.C., Marx, B.D.: Flexible smoothing using B-splines and penalties (with comments and rejoinder). Stat. Sci. 11, 89–121 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Fahrmeir, L., Kneib, T.: On the identification of trend and correlation in temporal and spatial regression. In: Shalab, Heumann, C. (eds.) Recent Advances in Linear Models and Related Areas. Springer, Berlin (2008)

    Google Scholar 

  • Fahrmeir, L., Kneib, T.: Propriety of posteriors in structured additive regression models: Theory and empirical evidence. J. Stat. Plan. Inference 139, 843–859 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Fahrmeir, L., Kneib, T., Lang, S.: Penalized structured additive regression: A Bayesian perspective. Stat. Sin. 14, 731–761 (2004)

    MATH  MathSciNet  Google Scholar 

  • Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348–1360 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Fernandez, C., Ley, E., Steel, M.: Benchmark priors for Bayesian model averaging. J. Econom. 100, 381–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Frühwirth-Schnatter, S., Tüchler, R.: Bayesian parsimonious covariance estimation for hierarchical linear mixed models. Stat. Comput. 139, 1–13 (2008)

    Article  Google Scholar 

  • Frühwirth-Schnatter, S., Wagner, H.: Auxiliary mixture sampling for parameter-driven models of time series of small counts with applications to state space modelling. Biometrika 93, 827–841 (2006)

    Article  MathSciNet  Google Scholar 

  • Frühwirth-Schnatter, S., Frühwirth, R., Held, L., Rue, H.: Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. Stat. Comput. (2009, to appear)

  • Fu, W.J.: Penalized regression: The bridge versus the LASSO. J. Comput. Graph. Stat. 7, 397–416 (1998)

    Article  Google Scholar 

  • George, E.I., McCulloch, R.E.: Variable selection via Gibbs sampling. J. Am. Stat. Soc. 88, 881–889 (1993)

    Article  Google Scholar 

  • George, E.I., McCulloch, R.E.: Stochastic search variable selection. In: Spiegelhalter, et al. (eds.) Markov Chain Monte Carlo in Practice. Chapman & Hall, London (1995)

    Google Scholar 

  • George, E.I., McCulloch, R.E.: Approaches for Bayesian variable selection. Stat. Sin. 7, 339–374 (1997)

    MATH  Google Scholar 

  • Geweke, J.: Variable selection an model comparison in regression. In: Berger, et al. (eds.) Bayesian Statistics 5, pp. 609–620. Oxford University Press, Oxford (1996)

    Google Scholar 

  • Goeman, J.J.: An Efficient Algorithm for L1-penalized Estimation. University Medical Center, Leiden (2007)

    Google Scholar 

  • Griffin, J.E., Brown, P.J.: Alternative prior distributions for variable selection with very many more variables than observations. University of Warwick, Department of Statistics, Technical report (2005)

  • Griffin, J.E., Brown, P.J.: Bayesian adaptive lassos with non-convex penalization. University of Warwick, Department of Statistics, Technical report (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  • Ishwaran, H., Rao, S.J.: Detecting differentially expressed genes in microarrays using Bayesian model selection. J. Am. Stat. Assoc. 462, 438–455 (2003)

    Article  MathSciNet  Google Scholar 

  • Ishwaran, H., Rao, S.J.: Spike and slab variable selection: frequentist and Bayesian strategies. Ann. Stat. 33, 730–773 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Jullion, A., Lambert, P.: Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models. Comput. Stat. Data Anal. 51, 2542–2558 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Kammann, E.E., Wand, M.P.: Geoadditive models. Appl. Stat. 52, 1–18 (2003)

    MATH  MathSciNet  Google Scholar 

  • Kinney, S.K., Dunson, D.B.: Fixed and random effects selection in linear and logistic models. Biometrics 63, 690–698 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kneib, T., Fahrmeir, L.: A mixed model approach for geoadditive hazard regression. Scand. J. Stat. 34, 207–228 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Kneib, T., Konrath, S., Fahrmeir, L.: High-dimensional structured additive regression models: Bayesian regularisation, smoothing and predictive performance. Technical Report No. 46, Department of Statistics, Ludwig-Maximilians-University Munich (2009)

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

    Article  MathSciNet  Google Scholar 

  • Konrath, S., Kneib, T., Fahrmeir, L.: Bayesian regularization and smoothing for hazard regression. Technical Report No. 35, Department of Statistics, Ludwig-Maximilians-University Munich (2008)

  • Lang, S., Brezger, A.: Bayesian P-Splines. J. Comput. Graph. Stat. 13, 183–212 (2004)

    Article  MathSciNet  Google Scholar 

  • Ley, E., Steel, M.: On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. J. Appl. Econom. 24, 651–674 (2009)

    Article  Google Scholar 

  • Li, Y., Lin, X., Müller, P.: Bayesian inference in semiparametric mixed models for longitudinal data. Biometrics (2009, to appear)

  • Liang, F., Paulo, R., Molina, G., Clyde, M.A., Berger, J.O.: Mixtures of g priors for Bayesian variable selection. J. Am. Stat. Assoc. 103, 410–423 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Müller, P., Quintana, F.A.: Nonparametric Bayesian data analysis. Stat. Sci. 19, 95–110 (2004)

    Article  MATH  Google Scholar 

  • Nychka, D.: Spatial-process estimates as smoothers. In: Schimek, M. (ed.) Smoothing and Regression: Approaches, Computation and Application. Wiley, New York (2000)

    Google Scholar 

  • Panagiotelis, A., Smith, M.: Bayesian identification, selection and estimation of semiparametric functions in high-dimensional additive models. J. Econom. 143, 291–316 (2008)

    Article  MathSciNet  Google Scholar 

  • Park, T., Casella, G.: The Bayesian lasso. J. Am. Stat. Assoc. 103, 681–686 (2008)

    Article  MATH  Google Scholar 

  • Rue, H., Held, L.: Gaussian Markov Random Fields. Theory and Applications. CRC/Chapman & Hall, London (2005)

    MATH  Google Scholar 

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

    Google Scholar 

  • Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric Regression. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  • Scheipl, F., Kneib, T.: Locally adaptive Bayesian P-splines with a normal-exponential-gamma prior. Comput. Stat. Data Anal. 53, 3533–3552 (2009)

    Article  Google Scholar 

  • Smith, M., Kohn, R.: Nonparametric regression using Bayesian variable selection. J. Econom. 75, 317–343 (1996)

    Article  MATH  Google Scholar 

  • Smith, M., Kohn, R.: Parsimonious covariance matrix estimation for longitudinal data. J. Am. Stat. Assoc. 97, 1141–1153 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Therneau, T.M., Grambsch, P.M.: Modeling Survival Data: Extending the Cox Model. Springer, New York (2000)

    MATH  Google Scholar 

  • Tibshirani, R.: The LASSO method for variable selection in the Cox model. Stat. Med. 16, 385–395 (1997)

    Article  Google Scholar 

  • Tüchler, R.: Bayesian variable selection for logistic models using auxiliary mixture sampling. J. Comput. Graph. Stat. 17, 76–94 (2008)

    Article  Google Scholar 

  • Wand, M.P.: Smoothing and mixed models. Comput. Stat. 18, 223–249 (2003)

    MATH  Google Scholar 

  • Wang, X., George, E.I.: Adaptive Bayesian criteria in variable selection for generalized linear models. Stat. Sin. 17, 667–690 (2007)

    MATH  MathSciNet  Google Scholar 

  • Zhao, Y., Staudenmayer, J., Coull, B.A., Wand, M.P.: General design Bayesian generalized linear mixed models. Stat. Sci. 21, 35–51 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B 67, 301–320 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Ludwig-Maximilians-University, Munich, Germany

    Ludwig Fahrmeir & Susanne Konrath

  2. Department of Mathematics, Carl von Ossietzky University, Oldenburg, Germany

    Thomas Kneib

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  1. Ludwig Fahrmeir
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  2. Thomas Kneib
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  3. Susanne Konrath
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Correspondence to Thomas Kneib.

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Fahrmeir, L., Kneib, T. & Konrath, S. Bayesian regularisation in structured additive regression: a unifying perspective on shrinkage, smoothing and predictor selection. Stat Comput 20, 203–219 (2010). https://doi.org/10.1007/s11222-009-9158-3

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  • DOI: https://doi.org/10.1007/s11222-009-9158-3