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Properties of the posterior distribution of a regression model based on Gaussian random fields

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Abstract

We consider the regression problem based on Gaussian processes. We assume that the prior distribution on the vector of parameters in the corresponding model of the covariance function is non-informative. Under this assumption, we prove the Bernstein-von Mises theorem that states that the posterior distribution on the parameters vector is close to the corresponding normal distribution. We show results of numerical experiments that indicate that our results apply in practically important cases.

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References

  1. Rasmussen, C.E. and Williams, C.K.I., Gaussian Processes for Machine Learning, Cambridge: MIT Press, 2006.

    MATH  Google Scholar 

  2. Chervonenkis, A.Ya., Chernova, S.S., and Zykova, T.V., Applications of Kernel Ridge Estimation to the Problem of Computing the Aerodynamical Characteristics of a Passenger Plane (in Comparison with Results Obtained with Artificial Neural Networks), Autom. Remote Control, 2011, vol. 72, no. 5, pp. 1061–1067.

    Article  MATH  Google Scholar 

  3. Forrester, A., Sobester, A., and Keane, A., Engineering Design via Surrogate Modelling: A Practical Guide, Chichester: Wiley, 2008.

    Book  Google Scholar 

  4. Panov, M.E., Burnaev, E.V., and Zaitsev, A.A., On Methods of Introducing Regularization in Regression Based on Gaussian processes, Tr. konf. “Matematicheskie metody raspoznavaniya obrazov-15” (Proc. Conf. “Mathematical Methods of Image Recognition-15”), 2011, pp. 142–145.

    Google Scholar 

  5. Kennedy, M.C. and Hagan, A.O., Bayesian Calibration of Computer Models, J. R. Statist. Soc., Ser. B (Stat. Met.), 2001, vol. 63, no. 3, pp. 425–464.

    Article  MATH  Google Scholar 

  6. Qian, P.Z.G. and Wu, C.F.G., Bayesian Hierarchical Modeling for Integrating Low-accuracy and High-accuracy Experiments, Technometrics, 2008, vol. 50, no. 2, pp. 192–204.

    Article  MathSciNet  Google Scholar 

  7. Kaufman, C.G., Schervish, M.J., and Nychka, D.W., Covariance Tapering for Likelihood-based Estimation in Large Spatial Data Sets, J. Am. Statist. Ass., 2008, vol. 103, no. 484, pp. 1545–1555.

    Article  MathSciNet  Google Scholar 

  8. Eidsvik, J., Finley, A.O., Banerjee, S., et al., Approximate Bayesian Inference for Large Spatial Datasets Using Predictive Process Models, Comput. Statist. Data Anal., 2011, vol. 56, no. 6, pp. 1362–1380.

    Article  MathSciNet  Google Scholar 

  9. Shaby, B. and Ruppert, D., Tapered Covariance: Bayesian Estimation and Asymptotics, J. Comput. Graphical Statist., 2012, vol. 21, no. 2, pp. 433–452.

    Article  MathSciNet  Google Scholar 

  10. Spokoiny, V., Parametric Estimation. Finite Sample Theory, Ann. Statist., 2012, vol. 40, no. 6, pp. 2877–2909.

    Article  MathSciNet  Google Scholar 

  11. Spokoiny, V., Bernstein-von Mises Theorem for Growing Parameter Dimension, preprint, 2013, arXiv: 1302.3430v2.

    Google Scholar 

  12. Kok, S., The Asymptotic Behaviour of the Maximum Likelihood Function of Kriging Approximations Using the Gaussian Correlation Function, Int. Conf. on Engin. Optimization (EngOpt 2012), Rio de Janeiro, Brazil, 2012.

    Google Scholar 

  13. Nagy, B., Loeppky, J.L., and Welch, W.J., Correlation Parameterization in Random Function Models to Improve Normal Approximation of the Likelihood or Posterior, Technical Report, Vancouver: Univ. of British Columbia, 2007.

    Google Scholar 

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Original Russian Text © A.A. Zaitsev, E.V. Burnaev, V.G. Spokoiny, 2013, published in Avtomatika i Telemekhanika, 2013, No. 10, pp. 55–67.

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Zaitsev, A.A., Burnaev, E.V. & Spokoiny, V.G. Properties of the posterior distribution of a regression model based on Gaussian random fields. Autom Remote Control 74, 1645–1655 (2013). https://doi.org/10.1134/S0005117913100056

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