Bayesian Conditional Density Filtering

We propose a Conditional Density Filtering (C-DF) algorithm for efficient online Bayesian inference. C-DF adapts MCMC sampling to the online setting, sampling from approximations to conditional posterior distributions obtained by propagating surrogate conditional sufficient statistics (a function of data and parameter estimates) as new data arrive. These quantities eliminate the need to store or process the entire dataset simultaneously and offer a number of desirable features. Often, these include a reduction in memory requirements and runtime and improved mixing, along with state-of-the-art parameter inference and prediction. These improvements are demonstrated through several illustrative examples including an application to high dimensional compressed regression. Finally, we show that C-DF samples converge to the target posterior distribution asymptotically as sampling proceeds and more data arrives.Comment: 41 pages, 7 figures, 12 table

A Computationally Efficient Projection-Based Approach for Spatial Generalized Linear Mixed Models

Inference for spatial generalized linear mixed models (SGLMMs) for high-dimensional non-Gaussian spatial data is computationally intensive. The computational challenge is due to the high-dimensional random effects and because Markov chain Monte Carlo (MCMC) algorithms for these models tend to be slow mixing. Moreover, spatial confounding inflates the variance of fixed effect (regression coefficient) estimates. Our approach addresses both the computational and confounding issues by replacing the high-dimensional spatial random effects with a reduced-dimensional representation based on random projections. Standard MCMC algorithms mix well and the reduced-dimensional setting speeds up computations per iteration. We show, via simulated examples, that Bayesian inference for this reduced-dimensional approach works well both in terms of inference as well as prediction, our methods also compare favorably to existing "reduced-rank" approaches. We also apply our methods to two real world data examples, one on bird count data and the other classifying rock types

Penalized additive regression for space-time data: a Bayesian perspective

We propose extensions of penalized spline generalized additive models for analysing space-time regression data and study them from a Bayesian perspective. Non-linear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spatial effects follow a Markov random field prior. This allows to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference can be performed either with full (FB) or empirical Bayes (EB) posterior analysis. FB inference using MCMC techniques is a slight extension of own previous work. For EB inference, a computationally efficient solution is developed on the basis of a generalized linear mixed model representation. The second approach can be viewed as posterior mode estimation and is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to smoothing parameters, are then estimated by using marginal likelihood. We carefully compare both inferential procedures in simulation studies and illustrate them through real data applications. The methodology is available in the open domain statistical package BayesX and as an S-plus/R function

A review of R-packages for random-intercept probit regression in small clusters

Generalized Linear Mixed Models (GLMMs) are widely used to model clustered categorical outcomes. To tackle the intractable integration over the random effects distributions, several approximation approaches have been developed for likelihood-based inference. As these seldom yield satisfactory results when analyzing binary outcomes from small clusters, estimation within the Structural Equation Modeling (SEM) framework is proposed as an alternative. We compare the performance of R-packages for random-intercept probit regression relying on: the Laplace approximation, adaptive Gaussian quadrature (AGQ), Penalized Quasi-Likelihood (PQL), an MCMC-implementation, and integrated nested Laplace approximation within the GLMM-framework, and a robust diagonally weighted least squares estimation within the SEM-framework. In terms of bias for the fixed and random effect estimators, SEM usually performs best for cluster size two, while AGQ prevails in terms of precision (mainly because of SEM's robust standard errors). As the cluster size increases, however, AGQ becomes the best choice for both bias and precision