Robust Modeling Using Non-Elliptically Contoured Multivariate t Distributions

Models based on multivariate t distributions are widely applied to analyze data with heavy tails. However, all the marginal distributions of the multivariate t distributions are restricted to have the same degrees of freedom, making these models unable to describe different marginal heavy-tailedness. We generalize the traditional multivariate t distributions to non-elliptically contoured multivariate t distributions, allowing for different marginal degrees of freedom. We apply the non-elliptically contoured multivariate t distributions to three widely-used models: the Heckman selection model with different degrees of freedom for selection and outcome equations, the multivariate Robit model with different degrees of freedom for marginal responses, and the linear mixed-effects model with different degrees of freedom for random effects and within-subject errors. Based on the Normal mixture representation of our t distribution, we propose efficient Bayesian inferential procedures for the model parameters based on data augmentation and parameter expansion. We show via simulation studies and real examples that the conclusions are sensitive to the existence of different marginal heavy-tailedness

Robust Bayesian inference via coarsening

The standard approach to Bayesian inference is based on the assumption that the distribution of the data belongs to the chosen model class. However, even a small violation of this assumption can have a large impact on the outcome of a Bayesian procedure. We introduce a simple, coherent approach to Bayesian inference that improves robustness to perturbations from the model: rather than condition on the data exactly, one conditions on a neighborhood of the empirical distribution. When using neighborhoods based on relative entropy estimates, the resulting "coarsened" posterior can be approximated by simply tempering the likelihood---that is, by raising it to a fractional power---thus, inference is often easily implemented with standard methods, and one can even obtain analytical solutions when using conjugate priors. Some theoretical properties are derived, and we illustrate the approach with real and simulated data, using mixture models, autoregressive models of unknown order, and variable selection in linear regression

Prior distributions for objective Bayesian analysis

We provide a review of prior distributions for objective Bayesian analysis. We start by examining some foundational issues and then organize our exposition into priors for: i) estimation or prediction; ii) model selection; iii) highdimensional models. With regard to i), we present some basic notions, and then move to more recent contributions on discrete parameter space, hierarchical models, nonparametric models, and penalizing complexity priors. Point ii) is the focus of this paper: it discusses principles for objective Bayesian model comparison, and singles out some major concepts for building priors, which are subsequently illustrated in some detail for the classic problem of variable selection in normal linear models. We also present some recent contributions in the area of objective priors on model space.With regard to point iii) we only provide a short summary of some default priors for high-dimensional models, a rapidly growing area of research