Stochastic substitution, the Gibbs sampler, and the sampling-importance-resampling algorithm can be viewed as three alternative sampling- (or Monte Carlo-) based approaches to the calculation of numerical estimates of marginal probability distributions. The three approaches will be reviewed, compared, and contrasted in relation to various joint probability structures frequently encountered in applications. In particular, the relevance of the approaches to calculating Bayesian posterior densities for a variety of structured models will be discussed and illustrated.

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Sampling-Based Approaches to Calculating Marginal Densities

Sampling-based approaches to calculating marginal densities

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. 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 owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling 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 is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

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Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

The idea of data augmentation arises naturally in missing value problems, as exemplified by the standard ways of filling in missing cells in balanced two-way tables. Thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze. This device is used to great advantage by the EM algorithm (Dempster, Laird, and Rubin 1977) in solving maximum likelihood problems. In situations when the likelihood cannot be approximated closely by the normal likelihood, maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements. From the Bayesian point of view, one must now calculate the posterior distribution of parameters of interest. If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution. It is the purpose of this article to explain how this can be done. The basic idea ...

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The calculation of posterior distributions by data augmentation

to the DLM: The First-Order Polynomial Model.- to the DLM: The Dynamic Regression Model.- The Dynamic Linear Model.- Univariate Time Series DLM Theory.- Model Specification and Design.- Polynomial Trend Models.- Seasonal Models.- Regression, Autoregression, and Related Models.- Illustrations and Extensions of Standard DLMs.- Intervention and Monitoring.- Multi-Process Models.- Non-Linear Dynamic Models: Analytic and Numerical Approximations.- Exponential Family Dynamic Models.- Simulation-Based Methods in Dynamic Models.- Multivariate Modelling and Forecasting.- Distribution Theory and Linear Algebra.

Bayesian Forecasting and Dynamic Models

Routine implementation of the Bayesian paradigm requires an efficient approach to the calculation and display of posterior or predictive distributions for given likelihood and prior specifi- cations. In this paper we shall review some of the analytic and numerical approaches currently available, describing in detail a numerical integration strategy based on Gaussian quadrature, and an associated strategy for the reconstruction and display of distributions based on spline techniques.

The implementation of the bayesian paradigm

One of the main obstacles to the routine implementation of Bayesian methods has been the absence of efficient algorithms for carrying out the computational tasks implicit in the Bayesian approach. In this paper, recent progress towards overcoming this problem is reviewed. In particular, novel numerical integration and interpolation methods, which exploit the opportunities offered by modern interactive computing and graphics facilities, are outlined and illustrated.

A. M. Skene

Papers

The implementation of the bayesian paradigm

Progress with numerical and graphical methods for practical Bayesian statistics