Generalized Additive Models (GAMs) via Bayesian P-splines using INLA

IFCS 2015 celebrated Generalized Additive Models (GAMs) with its back-fitting algorithm. With P-splines a direct fitting algorithm is feasible. With a prior on the smoothing parameter, Bayesian GAMs are obtained, which allow credible intervals that account for the uncertainty in the smoothing parameter. Integrated nested Laplace approximation (INLA) makes the Bayesian approach to GAMs practical and the INLA R package (Martins et al., 2013) already includes smoothing functions using piece-wise linear basis functions with peaks at the data points, supported by nice theory and links to geo-statistical modelling. Fitting Bayesian P-splines using INLA combines the advantages of the Bayesian approach using INLA with the flexibility and power of P-splines. Our approach maintains sparseness whereas the usual mixed model approach (Wakefield, 2013) does not. We are able to go beyond GAMs without increasing the number of penalty parameters beyond the number of covariates. We wrote a small R library (BayesianPspline) that wraps up the approach. We illustrate our approach using an ecological example (Fraaije et al., 2015). FRAAIJE, R. G. A., TER BRAAK, C. J. F., VERDUYN, B., BREEMAN, L. B. S., VERHOEVEN, J. T. A. & SOONS, M. B. (2015). Early plant recruitment stages set the template for the development of vegetation patterns along a hydrological gradient. Functional Ecology, http://dx.doi.org/10.1111/1365-2435.12441. MARTINS, T. G., SIMPSON, D., LINDGREN, F. & RUE, H. (2013). Bayesian computing with INLA: New features. Computational Statistics and Data Analysis, 67, 68-83. WAKEFIELD, J. (2013). Bayesian and Frequentist Regression Methods, edn. New York, NY, Springer New York.