  We define a novel class of additive models, called Extended Latent Gaussian
Models, that allow for a wide range of response distributions and flexible
relationships between the additive predictor and mean response. The new class
covers a broad range of interesting models including multi-resolution spatial
processes, partial likelihood-based survival models, and multivariate
measurement error models. Because computation of the exact posterior
distribution is infeasible, we develop a fast, scalable approximate Bayesian
inference methodology for this class based on nested Gaussian, Laplace, and
adaptive quadrature approximations. We prove that the error in these
approximate posteriors is op(1) under standard conditions, and provide
numerical evidence suggesting that our method runs faster and scales to larger
datasets than methods based on Integrated Nested Laplace Approximations and
Markov Chain Monte Carlo, with comparable accuracy. We apply the new method to
the mapping of malaria incidence rates in continuous space using aggregated
data, mapping leukaemia survival hazards using a Cox Proportional-Hazards model
with a continuously-varying spatial process, and estimating the mass of the
Milky Way Galaxy using noisy multivariate measurements of the positions and
velocities of star clusters in its orbit.
