Electronic Journal of Statistics

Improving the INLA approach for approximate Bayesian inference for latent Gaussian models

Egil Ferkingstad and Håvard Rue

Full-text: Open access

Abstract

We introduce a new copula-based correction for generalized linear mixed models (GLMMs) within the integrated nested Laplace approximation (INLA) approach for approximate Bayesian inference for latent Gaussian models. While INLA is usually very accurate, some (rather extreme) cases of GLMMs with e.g. binomial or Poisson data have been seen to be problematic. Inaccuracies can occur when there is a very low degree of smoothing or “borrowing strength” within the model, and we have therefore developed a correction aiming to push the boundaries of the applicability of INLA. Our new correction has been implemented as part of the R-INLA package, and adds only negligible computational cost. Empirical evaluations on both real and simulated data indicate that the method works well.

Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 2706-2731.

Dates
Received: April 2015
First available in Project Euclid: 14 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1450103327

Digital Object Identifier
doi:10.1214/15-EJS1092

Mathematical Reviews number (MathSciNet)
MR3433587

Zentralblatt MATH identifier
1329.62127

Subjects
Primary: 62F15: Bayesian inference

Keywords
Bayesian computation copulas generalized linear mixed models integrated nested Laplace approximation latent Gaussian models

Citation

Ferkingstad, Egil; Rue, Håvard. Improving the INLA approach for approximate Bayesian inference for latent Gaussian models. Electron. J. Statist. 9 (2015), no. 2, 2706--2731. doi:10.1214/15-EJS1092. https://projecteuclid.org/euclid.ejs/1450103327


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References

  • Azzalini, A. (2015). The R package sn: The skew-normal and skew-$t$ distributions (version 1.2-2), University of Padova, Italy, http://azzalini.stat.unipd.it/SN.
  • Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 579–602.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models., Journal of the American Statistical Association 88 9–25.
  • Capanu, M., Gönen, M. and Begg, C. B. (2013). An assessment of estimation methods for generalized linear mixed models with binary outcomes., Statistics in Medicine 32 4550–4566.
  • Cseke, B. and Heskes, T. (2010). Improving posterior marginal approximations in latent Gaussian models. In, International conference on Artificial Intelligence and Statistics 121–128.
  • De Backer, M., De Vroey, C., Lesaffre, E., Scheys, I. and De Keyser, P. (1998). Twelve weeks of continuous oral therapy for toenail onychomycosis caused by dermatophytes: A double-blind comparative trial of terbinafine 250 mg/day versus itraconazole 200 mg/day., Journal of the American Academy of Dermatology 38 S57–S63.
  • Evangelou, E., Zhu, Z. and Smith, R. L. (2011). Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation., Journal of Statistical Planning and Inference 141 3564–3577.
  • Fong, Y., Rue, H. and Wakefield, J. (2010). Bayesian inference for generalized linear mixed models., Biostatistics 11 397–412.
  • Grilli, L., Metelli, S. and Rampichini, C. (2014). Bayesian estimation with integrated nested Laplace approximation for binary logit mixed models., Journal of Statistical Computation and Simulation.
  • Illian, J. B., Sørbye, S. H. and Rue, H. (2012). A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA)., The Annals of Applied Statistics 6 1499–1530.
  • Kauermann, G., Krivobokova, T. and Fahrmeir, L. (2009). Some Asymptotic Results on Generalized Penalized Spline Smoothing., Journal of the Royal Statistical Society. Series B (Statistical Methodology) 71 487–503.
  • Lesaffre, E. and Spiessens, B. (2001). On the effect of the number of quadrature points in a logistic random effects model: An example., Journal of the Royal Statistical Society: Series C (Applied Statistics) 50 325–335.
  • Martins, T. G., Simpson, D., Lindgren, F. and Rue, H. (2013). Bayesian computing with INLA: New features., Computational Statistics & Data Analysis 67 68–83.
  • McCulloch, C. E., Searle, S. R. and Neuhaus, J. M. (2008)., Generalized, Linear, and Mixed Models, 2nd ed. John Wiley and sons, New York.
  • Nelsen, R. B. (2007)., An Introduction to Copulas, 2nd ed. Springer Science & Business Media, New York.
  • Ogden, H. E. (2015). A sequential reduction method for inference in generalized linear mixed models., Electronic Journal of Statistics 9 135–152.
  • Plummer, M. (2013). JAGS Version 3.4.0., http://mcmc-jags.sourceforge.net.
  • Raudenbush, S. W., Yang, M.-L. and Yosef, M. (2000). Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation., Journal of Computational and Graphical Statistics 9 141–157.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71 319–392.
  • Ruli, E., Sartori, N. and Ventura, L. (2015). Improved Laplace Approximation for Marginal Likelihoods., arXiv preprint arXiv:1502.06440.
  • Sauter, R. and Held, L. (2015). Quasi-complete Separation in Random Effects of Binary Response Mixed Models., Journal of Statistical Computation and Simulation. Accepted. DOI:10.1080/00949655.2015.1129539.
  • Shun, Z. and McCullagh, P. (1995). Laplace approximation of high dimensional integrals., Journal of the Royal Statistical Society. Series B (Methodological) 749–760.
  • Simpson, D., Illian, J., Lindgren, F., Sørbye, S. and Rue, H. (2013). Going off grid: Computationally efficient inference for log-Gaussian Cox processes., arXiv preprint arXiv:1111.0641.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and Van Der Linde, A. (2002). Bayesian measures of model complexity and fit., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64 583–639.