Bayes Linear Emulation, History Matching, and Forecasting for Complex Computer Simulators

  • Michael Goldstein
  • Nathan HuntleyEmail author
Reference work entry


Computer simulators are a useful tool for understanding complicated systems. However, any inferences made from them should recognize the inherent limitations and approximations in the simulator’s predictions for reality, the data used to run and calibrate the simulator, and the lack of knowledge about the best inputs to use for the simulator. This article describes the methods of emulation and history matching, where fast statistical approximations to the computer simulator (emulators) are constructed and used to reject implausible choices of input (history matching). Also described is a simple and tractable approach to estimating the discrepancy between simulator and reality induced by certain intrinsic limitations and uncertainties in the simulator and input data. Finally, a method for forecasting based on this approach is presented. The analysis is based on the Bayes linear approach to uncertainty quantification, which is similar in spirit to the standard Bayesian approach but takes expectation, rather than probability, as the primitive for the theory, with consequent simplifications in the prior uncertainty specification and analysis.


Computer simulators Bayes linear Emulation Model discrepancy History matching Calibration Internal discrepancy Forecasting 


  1. 1.
    Bastos, L.S., O’Hagan, A.: Diagnostics for Gaussian process emulators. Technometrics 51, 425–438 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clark, M.P., Slater, A.G., Rupp, D.E., Woods, R.A., Vrugt, J.A., Gupta, H.V., Wagener, T., Hay, L.E.: Framework for Understanding Structural Errors (FUSE): a modular framework to diagnose differences between hydrological models. Water Resour. Res. 44, W00B02 (2008)Google Scholar
  3. 3.
    Craig, P.S., Goldstein, M., Seheult, A.H., Smith, J.A.: Pressure matching for hydrocarbon reservoirs: a case study in the use of Bayes linear strategies for large computer experiments (with discussion). In: Gastonis, C., et al. (eds.) Case Studies in Bayesian Statistics, vol. III, pp. 37–93. Springer, New York (1997)CrossRefGoogle Scholar
  4. 4.
    Craig, P.S., Goldstein, M., Rougier, J.C., Seheult, A.H.: Bayesian forecasting using large computer models. JASA 96, 717–729 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cumming, J., Goldstein, M.: Small sample Bayesian designs for complex high-dimensional models based on information gained using fast approximations. Technometrics 51, 377–388 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    de Finetti, B.: Theory of Probability, vols. 1 & 2. Wiley, New York (1974, 1975)Google Scholar
  7. 7.
    Goldstein, M.: Subjective Bayesian analysis: principles and practice. Bayesian Anal. 1, 403–420 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goldstein, M., Rougier, J.C.: Bayes linear calibrated prediction for complex systems. JASA 101, 1132–1143 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldstein, M., Rougier, J.C.: Reified Bayesian modelling and inference for physical systems (with discussion). JSPI 139, 1221–1239 (2008)zbMATHGoogle Scholar
  10. 10.
    Goldstein, M., Seheult, A., Vernon, I.: Assessing model adequacy. In: Wainwright, J., Mulligan, M. (eds) Environmental Modelling: Finding Simplicity in Complexity, 2nd edn., pp. 435–449, Wiley, Chichester (2010)Google Scholar
  11. 11.
    Goldstein, M., Wooff, D.A.: Bayes Linear Statistics: Theory and Methods. Wiley, Chichester/Hoboken (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Monteith, J.L.: Evaporation and environment. Symp. Soc. Exp. Biol. 19, 205–224 (1965)Google Scholar
  13. 13.
    O’Hagan, A.: Bayesian analysis of computer code outputs: a tutorial. Reliab. Eng. Syst. Saf. 91, 1290–1300 (2006)CrossRefGoogle Scholar
  14. 14.
    Pukelsheim, F.: The three sigma rule. Am. Stat. 48, 88–91 (1994)MathSciNetGoogle Scholar
  15. 15.
    Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  16. 16.
    Vernon I., Goldstein M., and Bower, R.: Galaxy Formation: a Bayesian Uncertainty Analysis (with discussion). Bayesian Anal. 5, 619–670 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Williamson, D., Goldstein, M., Allison, L., Blaker, A., Challenor, P., Jackson, L., Yamazaki, K.: History matching for exploring and reducing climate model parameter space using observations and a large perturbed physics ensemble. Clim. Dyn. 41(7–8), 1703–1729 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Science Laboratories, Department of Mathematical SciencesDurham UniversityDurhamUK

Personalised recommendations