Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothing-spline models, state-space models, semiparametric regression, spatial and spatio-temporal models, log-Gaussian Cox-processes, geostatistical and geoadditive models. In this paper we consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form due to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, both in terms of convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations

Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geo-statistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the big-n problem, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is alltime-high, this fact seems still to be a computational bottleneck in applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the involved precision matrix sparse which enables the use of numerical algorithms for sparse matrices, that for fields in R 2 only use the square-root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parametrisation. In this paper, we show that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of R d, between GFs and GMRFs. The consequence is that we can take the best from the two worlds and do the modelling using GFs but do the computations using GMRFs. Perhaps more importantly,

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States of America.

An emulator is a statistical model of a deterministic function, to be used where the function itself is too expensive to evaluate within-the-loop of an inferential calculation. Typically, emulators are deployed when dealing with complex functions that have large and heterogeneous input and output spaces: environmental models, for example. In this challenging situation we should be sceptical about our statistical models, no matter how sophisticated, and adopt approaches that prioritise interpretative and diagnostic information, and the flexibility to respond. This paper presents one such approach, candidly rejecting the standard Smooth Gaussian Process approach in favour of a fully-Bayesian treatment of multivariate regression which, by permitting sequential updating, allows for very detailed predictive diagnostics. It is argued directly and by illustration that the incoherence of such a treatment (which does not impose continuity on the model outputs) is more than compensated for by the wealth of available information, and the possibilities for generalisation.

Likelihood-based methods such as maximum likelihood, REML, and Bayesian methods are attractive approaches to estimating covariance parameters in spatial models based on Gaussian processes. Finding such estimates can be computationally infeasible for large datasets, however, requiring O(n 3) calculations for each evaluation of the likelihood based on n observations. We propose the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrices are “tapered, ” or multiplied element-wise by a compactly supported correlation matrix. This produces matrices which can be be manipulated using more efficient sparse matrix algorithms. We present two approximations to the Gaussian likelihood using tapering. The first tapers the model covariance matrix only, whereas the second tapers both the model and sample covariance matrices. Tapering the model covariance matrix can be viewed as changing the underlying model to one in which the spatial covariance function is the direct product of the original covariance function and the tapering function. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which tapered and untapered covariance functions give equivalent (mutually absolutely continuous) measures for Gaussian processes on bounded domains. This allows us to evaluate

The deviance information criterion (DIC) is widely used for Bayesian model comparison, despite the lack of a clear theoretical foundation. DIC is shown to be an approximation to a penalized loss function based on the deviance, with a penalty derived from a cross-validation argument. This approximation is valid only when the effective number of parameters in the model is much smaller than the number of independent observations. In disease mapping, a typical application of DIC, this assumption does not hold and DIC under-penalizes more complex models. Another deviance-based loss function, derived from the same decision-theoretic framework, is applied to mixture models, which have previously been considered an unsuitable application for DIC.

We introduce a flexible parametric family of matrix-valued covariance functions for multivariate spatial random fields, where each constituent component is a Matérn process. The model parameters are interpretable in terms of process variance, smoothness, correlation length, and co-located correlation coefficients, which can be positive or negative. Both the marginal and the cross-covariance functions are of the Matérn type. In a data example on error fields for numerical predictions of surface pressure and temperature over the Pacific Northwest, a parsimonious bivariate Matérn model compares favorably to the traditional linear model of coregionalization.

Most studies of neighborhood effects on health have used the multilevel approach. However, since this meth-odology does not incorporate any notion of space, it may not provide optimal epidemiologic information when modeling variations or when investigating associations between contextual factors and health. Investigating mental disorders due to psychoactive substance use among all 65,830 individuals aged 40–59 years in 2001 in Malmö, Sweden, geolocated at their place of residence, the authors compared a spatial analytical perspective, which builds notions of space into hypotheses and methods, with the multilevel approach. Geoadditive models provided precise cartographic information on spatial variations in prevalence independent of administrative boundaries. The multi-level model showed significant neighborhood variations in the prevalence of substance-related disorders. However, hierarchical geostatistical models provided information on not only the magnitude but also the scale of neighbor-hood variations, indicating a significant correlation between neighborhoods in close proximity to each other. The prevalence of disorders increased with neighborhood deprivation. Far stronger associations were observed when using indicators measured in spatially adaptive areas, centered on residences of individuals, smaller in size than administrative neighborhoods. In neighborhood studies, building notions of space into analytical procedures may yield more comprehensive information than heretofore has been gathered on the spatial distribution of outcomes.