The key operation in Bayesian inference is to compute high-dimensional integrals. An old approximate technique is the Laplace method or approximation, which dates back to Pierre-Simon Laplace (1774). This simple idea approximates the integrand with a second-order Taylor expansion around the mode and computes the integral analytically. By developing a nested version of this classical idea, combined with modern numerical techniques for sparse matrices, we obtain the approach of integrated nested Laplace approximations (INLA) to do approximate Bayesian inference for latent Gaussian models (LGMs). LGMs represent an important model abstraction for Bayesian inference and include a large proportion of the statistical models used today. In this review, we discuss the reasons for the success of the INLA approach, the R-INLA package, why it is so accurate, why the approximations are very quick to compute, and why LGMs make such a useful concept for Bayesian computing.


Article metrics loading...

Loading full text...

Full text loading...


Literature Cited

  1. Azzalini A, Capitanio A. 1999. Statistical applications of the multivariate skew-normal distribution. J. R. Stat. Soc. B 61:579–602 [Google Scholar]
  2. Baghishani H, Mohammadzadeh M. 2012. Asymptotic normality of posterior distributions for generalized linear mixed models. J. Multivariate Anal. 111:66–77 [Google Scholar]
  3. Bakka H, Vanhatalo J, Illian J, Simpson D, Rue H. 2016. Accounting for physical barriers in species distribution modeling with non-stationary spatial random effects. arXiv1608.03787 [stat.AP]
  4. Barndorff-Nielsen OE, Cox DR. 1989. Asymptotic Techniques for Use in Statistics Boca Raton, FL: Chapman and Hall/CRC
  5. Bauer C, Wakefield J, Rue H, Self S, Feng Z, Wang Y. 2016. Bayesian penalized spline models for the analysis of spatio-temporal count data. Stat. Med. 35:1848–65 [Google Scholar]
  6. Bhatt S, Weiss DJ, Cameron E, Bisanzio D, Mappin B. et al. 2015. The effect of malaria control on Plasmodium falciparum in Africa between 2000 and 2015. Nature 526:207–11 [Google Scholar]
  7. Bivand RS, Gómez-Rubio V, Rue H. 2015. Spatial data analysis with R-INLA with some extensions. J. Stat. Softw. 63:1–31 [Google Scholar]
  8. Blangiardo M, Cameletti M. 2015. Spatial and Spatio-Temporal Bayesian Models with R-INLA New York: John Wiley & Sons
  9. Bolin D, Lindgren F. 2015. Excursion and contour uncertainty regions for latent Gaussian models. J. R. Stat. Soc. B 77:85–106 [Google Scholar]
  10. Bolin D, Lindgren F. 2016. Quantifying the uncertainty of contour maps. J. Comput. Graph. Stat. arXiv:1507.01778 [Google Scholar]
  11. Bowler DE, Haase P, Kröncke I, Tackenberg O, Bauer HG. et al. 2015. A cross-taxon analysis of the impact of climate change on abundance trends in central Europe. Biol. Conserv. 187:41–50 [Google Scholar]
  12. Box GEP, Tiao GC. 1973. Bayesian Inference in Statistical Analysis Reading, MA: Addison-Wesley
  13. Box GEP, Wilson KB. 1951. On the experimental attainment of optimum conditions (with discussion). J. R. Stat. Soc. B 13:1–45 [Google Scholar]
  14. Brown PE. 2015. Model-based geostatistics the easy way. J. Stat. Softw. 63:1–24 [Google Scholar]
  15. Crewe TL, Mccracken JD. 2015. Long-term trends in the number of monarch butterflies (Lepidoptera: Nymphalidae) counted on fall migration at Long Point, Ontario, Canada (1995–2014). Ann. Entomol. Soc. Am. 105:707–17 [Google Scholar]
  16. Dwyer-Lindgren L, Flaxman AD, Ng M, Hansen GM, Murray CJ, Mokdad AH. 2015. Drinking patterns in US counties from 2002 to 2012. Am. J. Public Health 105:1120–27 [Google Scholar]
  17. Ferkingstad E, Geirsson OP, Hrafnkelsson B, Davidsson OB, Gardarsson SM. 2016. A Bayesian hierarchical model for monthly maxima of instantaneous flow. arXiv1606.07667 [stat.AP]
  18. Ferkingstad E, Rue H. 2015. Improving the INLA approach for approximate Bayesian inference for latent Gaussian models. Electron. J. Stat. 9:2706–31 [Google Scholar]
  19. Friedrich A, Marshall JC, Biggs PJ, Midwinter AC, French NP. 2016. Seasonality of Campylobacter jejuni isolates associated with human campylobacteriosis in the Manawatu region, New Zealand. Epidemiol. Infect. 144:820–28 [Google Scholar]
  20. Fuglstad GA, Lindgren F, Simpson D, Rue H. 2015a. Exploring a new class of non-stationary spatial Gaussian random fields with varying local anisotropy. Stat. Sin. 25:115–33 [Google Scholar]
  21. Fuglstad GA, Simpson D, Lindgren F, Rue H. 2015b. Does non-stationary spatial data always require non-stationary random fields?. Spat. Stat. 14:C505–31 [Google Scholar]
  22. Fuglstad GA, Simpson D, Lindgren F, Rue H. 2016. Constructing priors that penalize the complexity of Gaussian random fields. arXiv1503.00256 [stat.ME]
  23. García-Pérez J, Lope V, López-Abente G, González-Sánchez M, Fernández-Navarro P. 2015. Ovarian cancer mortality and industrial pollution. Environ. Pollut. 205:103–10 [Google Scholar]
  24. Gelman A, Hwang J, Vehtari A. 2014. Understanding predictive information criteria for Bayesian models. Stat. Comput. 24:997–1016 [Google Scholar]
  25. Gneiting T, Raftery AE. 2007. Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc. 102:359–78 [Google Scholar]
  26. Goicoa T, Ugarte MD, Etxeberria J, Militino AF. 2016. Age-space-time CAR models in Bayesian disease mapping. Stat. Med. 35:2391–405 [Google Scholar]
  27. Goth US, Hammer HL, Claussen B. 2014. Utilization of Norway's emergency wards: the second 5 years after the introduction of the patient list system. Int. J. Environ. Res. Public Health 11:3375–86 [Google Scholar]
  28. Guihenneuc-Jouyaux C, Rousseau J. 2005. Laplace expansion in Markov chain Monte Carlo algorithms. J. Comput. Graphical Stat. 14:75–94 [Google Scholar]
  29. Guo J, Riebler A. 2015. Meta4diag: Bayesian bivariate meta-analysis of diagnostic test studies for routine practice. arXiv1512.06220 [stat.AP]
  30. Guo J, Rue H, Riebler A. 2015. Bayesian bivariate meta-analysis of diagnostic test studies with interpretable priors. arXiv1512.06217 [stat.ME]
  31. Halonen JI, Blangiardo M, Toledano MB, Fecht D, Gulliver J. et al. 2016. Long-term exposure to traffic pollution and hospital admissions in London. Environ. Pollut. 208:A48–57 [Google Scholar]
  32. Halonen JI, Hansell AL, Gulliver J, Morley D, Blangiardo M. et al. 2015. Road traffic noise is associated with increased cardiovascular morbidity and mortality and all-cause mortality in London. Eur. Heart J. 36:2653–61 [Google Scholar]
  33. Held L, Rue H. 2010. Conditional and intrinsic autoregressions. Handbook of Spatial Statistics A Gelfand, P Diggle, M Fuentes, P Guttorp 201–16 Boca Raton, FL: CRC/Chapman & Hall [Google Scholar]
  34. Held L, Sauter R. 2016. Adaptive prior weighting in generalized regression. Biometrics doi:10.1111/biom.12541
  35. Held L, Schrödle B, Rue H. 2010. Posterior and cross-validatory predictive checks: a comparison of MCMC and INLA. Statistical Modelling and Regression Structures—Festschrift in Honour of Ludwig Fahrmeir T Kneib, G Tutz 91–110 Berlin: Springer Verlag [Google Scholar]
  36. Henderson R, Shimakura S, Gorst D. 2002. Modeling spatial variation in leukemia survival data. J. Am. Stat. Assoc. 97:965–72 [Google Scholar]
  37. Hodges JS. 2013. Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects. Boca Raton, FL: Chapman and Hall/CRC
  38. Holand AM, Steinsland I, Martino S, Jensen H. 2013. Animal models and integrated nested Laplace approximations. G3 3:1241–51 [Google Scholar]
  39. Hu X, Steinsland I. 2016. Spatial modeling with system of stochastic partial differential equations. Wiley Interdiscip. Rev. Comput. Stat. 8:112–25 [Google Scholar]
  40. Iulian TV, Juan P, Mateu J. 2015. Bayesian spatio-temporal prediction of cancer dynamics. Comput. Math. Appl. 70:857–68 [Google Scholar]
  41. Jousimo J, Tack AJM, Ovaskainen O, Mononen T, Susi H. et al. 2014. Ecological and evolutionary effects of fragmentation on infectious disease dynamics. Science 344:1289–93 [Google Scholar]
  42. Kandt J, Chang S, Yip P, Burdett R. 2016. The spatial pattern of premature mortality in Hong Kong: How does it relate to public housing?. Urban Stud. doi: 10.1177/0042098015620341
  43. Karagiannis-Voules DA, Biedermann P, Ekpo UF, Garba A, Langer E. et al. 2015. Spatial and temporal distribution of soil-transmitted helminth infection in sub-Saharan Africa: a systematic review and geostatistical meta-analysis. Lancet Infect. Dis. 15:74–84 [Google Scholar]
  44. Karcher MD, Palacios JA, Bedford T, Suchard MA, Minin VN. 2016. Quantifying and mitigating the effect of preferential sampling on phylodynamic inference. PLOS Comput. Biol. 12:1–19 [Google Scholar]
  45. Kauermann G, Krivobokova T, Fahrmeir L. 2009. Some asymptotic results on generalized penalized spline smoothing. J. R. Stat. Soc. B 71:487–503 [Google Scholar]
  46. Klein N, Kneib T. 2016. Scale-dependent priors for variance parameters in structured additive distributional regression. Bayesian Anal. 11:1071–1106 [Google Scholar]
  47. Kröger H, Hoffmann R, Pakpahan E. 2016. Consequences of measurement error for inference in cross-lagged panel design—the example of the reciprocal causal relationship between subjective health and socio-economic status. J. R. Stat. Soc. A 179:607–28 [Google Scholar]
  48. Li Y, Brown P, Rue H, al-Maini M, Fortin P. 2012. Spatial modelling of lupus incidence over 40 years with changes in census areas. J. R. Stat. Soc. C 61:99–115 [Google Scholar]
  49. Lindgren F, Rue H. 2008. A note on the second order random walk model for irregular locations. Scand. J. Stat. 35:691–700 [Google Scholar]
  50. Lindgren F, Rue H. 2015. Bayesian spatial modelling with R-INLA. J. Stat. Softw. 63:1–25 [Google Scholar]
  51. 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. B 73:423–98 [Google Scholar]
  52. Lithio A, Nettleton D. 2015. Hierarchical modeling and differential expression analysis for RNA-seq experiments with inbred and hybrid genotypes. J. Agric. Biol. Environ. Stat. 20:598–613 [Google Scholar]
  53. Martino S, Akerkar R, Rue H. 2010. Approximate Bayesian inference for survival models. Scand. J. Stat. 28:514–28 [Google Scholar]
  54. Martins TG, Rue H. 2014. Extending INLA to a class of near-Gaussian latent models. Scand. J. Stat. 41:893–912 [Google Scholar]
  55. Martins TG, Simpson D, Lindgren F, Rue H. 2013. Bayesian computing with INLA: new features. Comput. Stat. Data Anal. 67:68–83 [Google Scholar]
  56. Muff S, Keller LF. 2015. Reverse attenuation in interaction terms due to covariate measurement error. Biometrical J. 57:1068–83 [Google Scholar]
  57. Muff S, Riebler A, Rue H, Saner P, Held L. 2015. Bayesian analysis of measurement error models using integrated nested Laplace approximations. J. R. Stat. Soc. C 64:231–52 [Google Scholar]
  58. Niemi J, Mittman E, Landau W, Nettleton D. 2015. Empirical Bayes analysis of RNA-seq data for detection of gene expression heterosis. J. Agric. Biol. Environ. Stat. 20:614–28 [Google Scholar]
  59. Noor AM, Kinyoki DK, Mundia CW, Kabaria CW, Mutua JW. et al. 2014. The changing risk of Plasmodium falciparum malaria infection in Africa: 2000-10: a spatial and temporal analysis of transmission intensity. Lancet 383:1739–47 [Google Scholar]
  60. Ogden H. 2016. On asymptotic validity of approximate likelihood inference. arXiv1601.07911 [math.ST]
  61. Opitz N, Marcon C, Paschold A, Malik WA, Lithio A. et al. 2016. Extensive tissue-specific transcriptomic plasticity in maize primary roots upon water deficit. J. Exp. Bot. 67:1095–107 [Google Scholar]
  62. Papoila AL, Riebler A, Amaral-Turkman A, São-João R, Ribeiro C. et al. 2014. Stomach cancer incidence in Southern Portugal 1998–2006: a spatio-temporal analysis. Biometrical J. 56:403–15 [Google Scholar]
  63. Plummer M. 2016. Rjags: Bayesian graphical models using MCMC. R Software Package for Graphical Models. https://cran.r-project.org/web/packages/rjags/index.html
  64. Quiroz Z, Prates MO, Rue H. 2015. A Bayesian approach to estimate the biomass of anchovies in the coast of Perú. Biometrics 71:208–17 [Google Scholar]
  65. Riebler A, Held L. 2016. Projecting the future burden of cancer: Bayesian age-period-cohort analysis with integrated nested Laplace approximations. Biometrical J. In press
  66. Riebler A, Held L, Rue H. 2012. Estimation and extrapolation of time trends in registry data—borrowing strength from related populations. Ann. Appl. Stat. 6:304–33 [Google Scholar]
  67. Riebler A, Robinson M, van de Wiel M. 2014. Analysis of next generation sequencing data using integrated nested Laplace approximation (INLA). Statistical Analysis of Next Generation Sequencing Data S Datta, D Nettleton 75–91 New York: Springer [Google Scholar]
  68. Riebler A, Sørbye SH, Simpson D, Rue H. 2016. An intuitive Bayesian spatial model for disease mapping that accounts for scaling. Stat. Methods Med. Res. 25:1145–65 [Google Scholar]
  69. Robert CP, Casella G. 1999. Monte Carlo Statistical Methods New York: Springer-Verlag
  70. Rooney J, Vajda A, Heverin M, Elamin M, Crampsie A. et al. 2015. Spatial cluster analysis of population amyotrophic lateral sclerosis risk in Ireland. Neurology 84:1537–44 [Google Scholar]
  71. Roos M, Held L. 2011. Sensitivity analysis in Bayesian generalized linear mixed models for binary data. Bayesian Anal. 6:259–78 [Google Scholar]
  72. Roos M, Martins TG, Held L, Rue H. 2015. Sensitivity analysis for Bayesian hierarchical models. Bayesian Anal. 10:321–49 [Google Scholar]
  73. Roos NC, Carvalho AR, Lopes PF, Pennino MG. 2015. Modeling sensitive parrotfish (Labridae: Scarini) habitats along the Brazilian coast. Mar. Environ. Res. 110:92–100 [Google Scholar]
  74. Rue H, Held L. 2005. Gaussian Markov Random Fields: Theory and Applications Boca Raton, FL: CRC/Chapman and Hall
  75. Rue H, Held L. 2010. Markov random fields. Handbook of Spatial Statistics A Gelfand, P Diggle, M Fuentes, P Guttorp 171–200 Boca Raton, FL: CRC/Chapman and Hall [Google Scholar]
  76. Rue H, Martino S, Chopin N. 2009. Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations (with discussion). J. R. Stat. Soc. B 71:319–92 [Google Scholar]
  77. Salmon M, Schumacher D, Stark K, Höhle M. 2015. Bayesian outbreak detection in the presence of reporting delays. Biometrical J. 57:1051–67 [Google Scholar]
  78. Santermans E, Robesyn E, Ganyani T, Sudre B, Faes C. et al. 2016. Spatiotemporal evolution of Ebola virus disease at sub-national level during the 2014 West Africa epidemic: model scrutiny and data meagreness. PLOS ONE 11:e0147172 [Google Scholar]
  79. Sauter R, Held L. 2015. Network meta-analysis with integrated nested Laplace approximations. Biometrical J. 57:1038–50 [Google Scholar]
  80. Sauter R, Held L. 2016. Quasi-complete separation in random effects of binary response mixed models. J. Stat. Comput. Simul. 86:2781–96 [Google Scholar]
  81. Schrödle B, Held L. 2011a. A primer on disease mapping and ecological regression using INLA. Comput. Stat. 26:241–58 [Google Scholar]
  82. Schrödle B, Held L. 2011b. Spatio-temporal disease mapping using INLA. Environmetrics 22:725–34 [Google Scholar]
  83. Schrödle B, Held L, Rue H. 2012. Assessing the impact of network data on the spatio-temporal spread of infectious diseases. Biometrics 68:736–44 [Google Scholar]
  84. Selwood KE, Thomson JR, Clarke RH, McGeoch MA, Mac Nally R. 2015. Resistance and resilience of terrestrial birds in drying climates: Do floodplains provide drought refugia?. Glob. Ecol. Biogeogr. 24:838–48 [Google Scholar]
  85. Shun Z, McCullagh P. 1995. Laplace approximation of high dimensional integrals. J. R. Stat. Soc. B 57:749–60 [Google Scholar]
  86. Simpson D, Illian J, Lindgren F, Sørbye S, Rue H. 2016a. Going off grid: computational efficient inference for log-Gaussian Cox processes. Biometrika 103:1–22 [Google Scholar]
  87. Simpson DP, Lindgren F, Rue H. 2011. Think continuous: Markovian Gaussian models in spatial statistics. Spat. Stat. 1:16–29 [Google Scholar]
  88. Simpson D, Lindgren F, Rue H. 2012. In order to make spatial statistics computationally feasible, we need to forget about the covariance function. Environmetrics 23:65–74 [Google Scholar]
  89. Simpson DP, Rue H, Riebler A, Martins TG, Sørbye SH. 2016b. Penalising model component complexity: a principled, practical approach to constructing priors (with discussion). Stat. Sci. In press
  90. Sørbye SH, Rue H. 2014. Scaling intrinsic Gaussian Markov random field priors in spatial modelling. Spat. Stat. 8:39–51 [Google Scholar]
  91. Sørbye SH, Rue H. 2016. Penalised complexity priors for stationary autoregressive processes. arXiv1608.08941 [stat.ME]
  92. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A. 2002. Bayesian measures of model complexity and fit (with discussion). J. R. Stat. Soc. B 64:583–639 [Google Scholar]
  93. Spiegelhalter DJ, Thomas A, Best NG, Gilks WR. 1995. BUGS: Bayesian inference using Gibbs sampling. Software Package for Performing Bayesian Inference Using Markov Chain Monte Carlo. http://www.mrc-bsu.cam.ac.uk/software/bugs/
  94. Stan Development Team 2015. Stan modeling language user's guide and reference manual. http://www.uvm.edu/∼bbeckage/Teaching/DataAnalysis/Manuals/stan-reference-2.8.0.pdf
  95. Tierney L, Kadane JB. 1986. Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81:82–86 [Google Scholar]
  96. Tsiko RG. 2016. A spatial latent Gaussian model for intimate partner violence against men in Africa. J. Fam. Violence 41:443–59 [Google Scholar]
  97. Ugarte MD, Adin A, Goicoa T. 2016. Two-level spatially structured models in spatio-temporal disease mapping. Stat. Methods Med. Res. 25:1080–100 [Google Scholar]
  98. Ugarte MD, Adin A, Goicoa T, Militino AF. 2014. On fitting spatio-temporal disease mapping models using approximate Bayesian inference. Stat. Methods Med. Res. 23:507–30 [Google Scholar]
  99. Van De Wiel MA, De Menezes RX, Siebring E, Van Beusechem VW. 2013a. Analysis of small-sample clinical genomics studies using multi-parameter shrinkage: application to high-throughput RNA interference screening. BMC Med. Genom. 6:1–9 [Google Scholar]
  100. Van De Wiel MA, Leday GGR, Pardo L, Rue H, van der Vaart AW, van Wieringen WN. 2013b. Bayesian analysis of high-dimensional RNA sequencing data: estimating priors for shrinkage and multiplicity correction. Biostatistics 14:113–28 [Google Scholar]
  101. Van De Wiel MA, Neerincx M, Buffart TE, Sie D, Verheul HMW. 2014. ShrinkBayes: a versatile R-package for analysis of count-based sequencing data in complex study design. BMC Bioinform. 15:116 [Google Scholar]
  102. Ventrucci M, Rue H. 2016. Penalized complexity priors for degrees of freedom in Bayesian P-splines. Stat. Model. doi:10.1177/1471082X16659154. In press
  103. Wantanabe S. 2010. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J. Mach. Learn. Res. 11:3571–94 [Google Scholar]
  104. Whittle P. 1954. On stationary processes in the plane. Biometrika 41:434–49 [Google Scholar]
  105. Whittle P. 1963. Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40:974–94 [Google Scholar]
  106. Yuan Y, Bachl FE, Borchers DL, Lindgren F, Illian JB. et al. 2016. Point process models for spatio-temporal distance sampling data. arXiv1604.06013 [stat.ME]
  107. Yue YR, Simpson D, Lindgren F, Rue H. 2014. Bayesian adaptive smoothing spline using stochastic differential equations. Bayesian Anal. 9:397–424 [Google Scholar]

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error