Simplified integrated nested Laplace approximation

Summary Integrated nested Laplace approximation provides accurate and efficient approximations for marginal distributions in latent Gaussian random field models. Computational feasibility of the original Rue et al. (2009) methods relies on efficient approximation of Laplace approximations for the marginal distributions of the coefficients of the latent field, conditional on the data and hyperparameters. The computational efficiency of these approximations depends on the Gaussian field having a Markov structure. This note provides equivalent efficiency without requiring the Markov property, which allows for straightforward use of latent Gaussian fields without a sparse structure, such as reduced rank multi-dimensional smoothing splines. The method avoids the approximation for conditional modes used in Rue et al. (2009), and uses a log determinant approximation based on a simple quasi-Newton update. The latter has a desirable property not shared by the most commonly used variant of the original method.