Spatial Models Using Laplace Approximation Methods

  • Virgilio Gómez-Rubio
  • Roger S. Bivand
  • Håvard Rue
Reference work entry


Bayesian inference has been at the center of the development of spatial statistics in recent years. In particular, Bayesian hierarchical models including several fixed and random effects have become very popular in many different fields. Given that inference on these models is seldom available in closed form, model fitting is usually based on simulation methods such as Markov chain Monte Carlo.

However, these methods are often very computationally expensive and a number of approximations have been developed. The integrated nested Laplace approximation (INLA) provides a general approach to computing the posterior marginals of the parameters in the model. INLA focuses on latent Gaussian models, but this is a class of methods wide enough to tackle a large number of problems in spatial statistics.

In this chapter, we describe the main advantages of the integrated nested Laplace approximation. Applications to many different problems in spatial statistics will be discussed as well.


Posterior Distribution Gaussian Approximation Stochastic Partial Differential Equation Bayesian Hierarchical Model Laplace Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Virgilio Gómez-Rubio has been supported by the Spanish Ministry of Science and Innovation (project MTM 2008–03085) and Junta de Comunidades de Castilla–La Mancha (project PPIC11-0183-7474).


  1. Banerjee S, Gelfand AE, Carlin BP (2004) Hierarchical modeling and analysis for spatial data. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  2. Besag J, York J, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43(1):1–59CrossRefGoogle Scholar
  3. Bivand RS, Pebesma EJ, Gómez-Rubio V (2008) Applied spatial data analysis with R. Springer, New YorkGoogle Scholar
  4. Cameletti M, Lindgren F, Simpson D, Rue H (2012) Spatio-temporal modeling of particulate matter concentration through the SPDE approach. Adv Statistical Anal.
  5. Diggle P, Ribeiro PJ (2007) Model-based geostatistics. Springer, New YorkGoogle Scholar
  6. Diggle PJ, Menezes R, Tl S (2010) Geostatistical inference under preferential sampling. J R Stat Soc Ser C Appl Stat 59(2):191–232CrossRefGoogle Scholar
  7. Eidsvik J, Martino S, Rue H (2009) Approximate Bayesian inference in spatial generalized linear mixed models. Scand J Stat 36(1):1–22Google Scholar
  8. Fahrmeir L, Kneib T (2011) Bayesian smoothing and regression for longitudinal. Spatial and event history data. Oxford University Press, New YorkCrossRefGoogle Scholar
  9. Fuglstad GA (2011) Spatial modelling and inference with SPDE-based GMRFs. Master’s thesis, Norwegian University of Science and Technology, NorwayGoogle Scholar
  10. Geisser S (1993) Predictive inference: an introduction. Chapman & Hall, New YorkCrossRefGoogle Scholar
  11. Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis, 2nd edn. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  12. Held L, Schödle B, Rue H (2010) Posterior and cross-validatory predictive checks: a comparison of MCMC and INLA. In: Kneib T, Tutz G (eds) Statistical modelling and regression structures – Festschrift in Honour of Ludwig Fahrmeir. Springer, Berlin, pp 91–110CrossRefGoogle Scholar
  13. Illian JB, Martino S, Sorbye S, Gallego-Fernandez J, Travis J (2012) Fitting complex ecological point processes with integrated nested Laplace approximation (inla). Methods Ecol Evol.
  14. Kammann EE, Wand MP (2003) Geoadditive models. J R Stat Soc Ser C Appl Stat 52(1):1–18CrossRefGoogle Scholar
  15. Knorr-Held L (2000) Bayesian modelling of inseparable space-time variation in disease risk. Stat Med 19:2555–2567CrossRefGoogle Scholar
  16. Lang S, Brezger A (2004) Bayesian p-splines. J Comput Graph Stat 13(1):183–212CrossRefGoogle Scholar
  17. Lee DJ, Durbán M (2009) Smooth-car mixed models for spatial count data. Comput Stat Data Anal 53(8):2968–2979CrossRefGoogle Scholar
  18. Lindgren F, Rue H, Lindström J (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the SPDE approach (with discussion). J R Stat Soc Ser B 73(4):423–498CrossRefGoogle Scholar
  19. Marshall EC, Spiegelhalter DJ (2003) Approximate cross-validatory predictive checks in disease mapping models. Stat Med 22(10):1649–1660CrossRefGoogle Scholar
  20. Martino S, Rue H (2010) Case studies in Bayesian computation using INLA. In: Mantovan P, Secchi P (eds) Complex data modeling and computationally intensive statistical methods, Contributions to statistics. Springer, New York, pp 99–114CrossRefGoogle Scholar
  21. Pettit LI (1990) The conditional predictive ordinate for the normal distribution. J R Stat Soc Ser B Methodol 52(1):175–184Google Scholar
  22. R Development Core Team (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. ISBN 3-900051-07-0
  23. Roos M, Held L (2011) Sensitivity analysis in Bayesian generalized linear mixed models for binary data. Bayesian Anal 6(2):259–278CrossRefGoogle Scholar
  24. Rue H, Held L (2005) Gaussian Markov random fields. Theory and applications. Chapman & Hall, New YorkCrossRefGoogle Scholar
  25. Rue H, Martino S (2007) Approximate Bayesian inference for hierarchical Gaussian Markov random field models. J Stat Plan Inference 137(10, SI):3177–3192CrossRefGoogle Scholar
  26. Rue H, Martino S, Chopin N (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J R Stat Soc Ser B Stat Methodol 71(Pt 2):319–392CrossRefGoogle Scholar
  27. Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, New YorkCrossRefGoogle Scholar
  28. Schroedle B, Held L, Riebler A, Danuser J (2011) Using integrated nested Laplace approximations for the evaluation of veterinary surveillance data from Switzerland: a case-study. J R Stat Soc Ser C Appl Stat 60(Pt 2):261–279CrossRefGoogle Scholar
  29. Simpson D, Illian J, Lindgren F, Sørbye SH, Rue H (2011) Going off grid: computationally efficient inference for log-Gaussian Cox processes. Preprint Statistics 10/2011. Norwegian University of Science and Technology, TrondheimGoogle Scholar
  30. Spiegelhalter DJ, Best NG, Carlin BP, Van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). J R Stat Soc Ser B 64(4):583–616CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Virgilio Gómez-Rubio
    • 1
  • Roger S. Bivand
    • 2
  • Håvard Rue
    • 3
  1. 1.Department of Mathematics, School of Industrial Engineering-AlbaceteUniversity of Castilla-La ManchaAlbaceteSpain
  2. 2.Department of EconomicsNHH Norwegian School of EconomicsBergenNorway
  3. 3.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations