Abstract
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.
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References
Baddeley A, Rubak E, Turner R (2015) Spatial point patterns: methodology and applications with R. Chapman & Hall/CRC Press, London. http://www.crcpress.com/Spatial-Point-Patterns-Methodology-and-Applications-with-R/Baddeley-Rubak-Turner/9781482210200/
Bakka H, Rue H, Fuglstad GA, Riebler A, Bolin D, Krainski E, Simpson D, Lindgren F (2018) Spatial modelling with R-INLA: a review. WIREs Comput Stat 10(6):1–24. https://doi.org/10.1002/wics.1443
Banerjee S, Gelfand AE, Carlin BP (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC, Boca Raton
Besag J, York J, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43(1):1–59
Diggle PJ, Menezes R, Su T (2010) Geostatistical inference under preferential sampling. J R Stat Soc Ser C Appl Stat 59(2):191–232
Fahrmeir L, Kneib T (2011) Bayesian smoothing and regression for longitudinal. Spatial and event history data. Oxford University Press, New York
Geisser S (1993) Predictive inference: an introduction. Chapman & Hall, New York
Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis, 2nd edn. Chapman & Hall/CRC, Boca Raton
Kammann EE, Wand MP (2003) Geoadditive models. J R Stat Soc Ser C Appl Stat 52(1):1–18
Lang S, Brezger A (2004) Bayesian P-splines. J Comput Graph Stat 13(1):183–212
Lee DJ, Durbán M (2009) Smooth-car mixed models for spatial count data. Comput Stat Data Anal 53(8):2968–2979
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):423–498
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 99–114
Pettit LI (1990) The conditional predictive ordinate for the normal distribution. J R Stat Soc Ser B Methodol 52(1):175–184
Roos M, Held L (2011) Sensitivity analysis in Bayesian generalized linear mixed models for binary data. Bayesian Anal 6(2):259–278
Rue H, Held L (2005) Gaussian Markov random fields. Theory and applications. Chapman & Hall, New York
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 71(Part 2):319–392
Rue H, Riebler AI, Sørbye SH, Illian JB, Simpson DP, Lindgren FK (2017) Bayesian computing with INLA: a review. Ann Rev Stat Appl 4:395–421
Simpson D, Illian J, Lindgren F, Sørbye SH, Rue H (2016) Going off grid: computationally efficient inference for log-gaussian cox processes. Biometrika 103(1):49–70
Acknowledgements
V. Gómez-Rubio has been supported by grants PPIC-2014-001-P and SBPLY/17/180501/000491, funded by Consejería de Educación, Cultura y Deportes (JCCM, Spain) and Fondo Europeo de Desarrollo Regional, and grants MTM2008-03085 and MTM2016-77501-P, funded by the Ministerio de Economía y Competitividad (Spain).
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Gómez-Rubio, V., Bivand, R.S., Rue, H. (2019). Spatial Models Using Laplace Approximation Methods. In: Fischer, M., Nijkamp, P. (eds) Handbook of Regional Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36203-3_104-1
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DOI: https://doi.org/10.1007/978-3-642-36203-3_104-1
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