The Inverse G-Wishart Distribution and Variational Message Passing

Message passing on a factor graph is a powerful paradigm for the coding of approximate inference algorithms for arbitrarily graphical large models. The notion of a factor graph fragment allows for compartmentalization of algebra and computer code. We show that the Inverse G-Wishart family of distributions enables fundamental variational message passing factor graph fragments to be expressed elegantly and succinctly. Such fragments arise in models for which approximate inference concerning covariance matrix or variance parameters is made, and are ubiquitous in contemporary statistics and machine learning

Asymptotics and optimal bandwidth selection for highest density region estimation

We study kernel estimation of highest-density regions (HDR). Our main contributions are two-fold. First, we derive a uniform-in-bandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties. We also present the results of numerical studies that illustrate the benefits of our theory and methodology.Comment: Published in at http://dx.doi.org/10.1214/09-AOS766 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Smoothing with Mixed Model Software

Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such as additive and varying coefficient models for both S-PLUS and SAS software. Code for each of the illustrations is available on the Internet.

Bayesian Analysis for Penalized Spline Regression Using WinBUGS

Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. Good mixing properties of the MCMC chains are obtained by using low-rank thin-plate splines, while simulation times per iteration are reduced employing WinBUGS specific computational tricks.

General Design Bayesian Generalized Linear Mixed Models

Linear mixed models are able to handle an extraordinary range of complications in regression-type analyses. Their most common use is to account for within-subject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when treated in full generality, mixed models can also handle spline-type smoothing and closely approximate kriging. This allows for nonparametric regression models (e.g., additive models and varying coefficient models) to be handled within the mixed model framework. The key is to allow the random effects design matrix to have general structure; hence our label general design. For continuous response data, particularly when Gaussianity of the response is reasonably assumed, computation is now quite mature and supported by the R, SAS and S-PLUS packages. Such is not the case for binary and count responses, where generalized linear mixed models (GLMMs) are required, but are hindered by the presence of intractable multivariate integrals. Software known to us supports special cases of the GLMM (e.g., PROC NLMIXED in SAS or glmmML in R) or relies on the sometimes crude Laplace-type approximation of integrals (e.g., the SAS macro glimmix or glmmPQL in R). This paper describes the fitting of general design generalized linear mixed models. A Bayesian approach is taken and Markov chain Monte Carlo (MCMC) is used for estimation and inference. In this generalized setting, MCMC requires sampling from nonstandard distributions. In this article, we demonstrate that the MCMC package WinBUGS facilitates sound fitting of general design Bayesian generalized linear mixed models in practice.Comment: Published at http://dx.doi.org/10.1214/088342306000000015 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Theory of Gaussian variational approximation for a Poisson mixed model

Likelihood-based inference for the parameters of generalized linear mixed models is hindered by the presence of intractable integrals. Gaussian variational approximation provides a fast and effective means of approximate inference. We provide some theory for this type of approximation for a simple Poisson mixed model. In particular, we establish consistency at rate m -1/2 +n-1, where m is the number of groups and n is the number of repeated measurements