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

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

Latent Gaussian modeling and INLA: A review with focus on space-time applications

Bayesian hierarchical models with latent Gaussian layers have proven very flexible in capturing complex stochastic behavior and hierarchical structures in high-dimensional spatial and spatio-temporal data. Whereas simulation-based Bayesian inference through Markov Chain Monte Carlo may be hampered by slow convergence and numerical instabilities, the inferential framework of Integrated Nested Laplace Approximation (INLA) is capable to provide accurate and relatively fast analytical approximations to posterior quantities of interest. It heavily relies on the use of Gauss-Markov dependence structures to avoid the numerical bottleneck of high-dimensional nonsparse matrix computations. With a view towards space-time applications, we here review the principal theoretical concepts, model classes and inference tools within the INLA framework. Important elements to construct space-time models are certain spatial Mat\'ern-like Gauss-Markov random fields, obtained as approximate solutions to a stochastic partial differential equation. Efficient implementation of statistical inference tools for a large variety of models is available through the INLA package of the R software. To showcase the practical use of R-INLA and to illustrate its principal commands and syntax, a comprehensive simulation experiment is presented using simulated non Gaussian space-time count data with a first-order autoregressive dependence structure in time

Projection predictive model selection for Gaussian processes

We propose a new method for simplification of Gaussian process (GP) models by projecting the information contained in the full encompassing model and selecting a reduced number of variables based on their predictive relevance. Our results on synthetic and real world datasets show that the proposed method improves the assessment of variable relevance compared to the automatic relevance determination (ARD) via the length-scale parameters. We expect the method to be useful for improving explainability of the models, reducing the future measurement costs and reducing the computation time for making new predictions.Comment: A few minor changes in tex

Understanding and Comparing Scalable Gaussian Process Regression for Big Data

As a non-parametric Bayesian model which produces informative predictive distribution, Gaussian process (GP) has been widely used in various fields, like regression, classification and optimization. The cubic complexity of standard GP however leads to poor scalability, which poses challenges in the era of big data. Hence, various scalable GPs have been developed in the literature in order to improve the scalability while retaining desirable prediction accuracy. This paper devotes to investigating the methodological characteristics and performance of representative global and local scalable GPs including sparse approximations and local aggregations from four main perspectives: scalability, capability, controllability and robustness. The numerical experiments on two toy examples and five real-world datasets with up to 250K points offer the following findings. In terms of scalability, most of the scalable GPs own a time complexity that is linear to the training size. In terms of capability, the sparse approximations capture the long-term spatial correlations, the local aggregations capture the local patterns but suffer from over-fitting in some scenarios. In terms of controllability, we could improve the performance of sparse approximations by simply increasing the inducing size. But this is not the case for local aggregations. In terms of robustness, local aggregations are robust to various initializations of hyperparameters due to the local attention mechanism. Finally, we highlight that the proper hybrid of global and local scalable GPs may be a promising way to improve both the model capability and scalability for big data.Comment: 25 pages, 15 figures, preprint submitted to KB

Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression

We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation method in practice