Bayesian multivariate mixed-scale density estimation

Although continuous density estimation has received abundant attention in the Bayesian nonparametrics literature, there is limited theory on multivariate mixed scale density estimation. In this note, we consider a general framework to jointly model continuous, count and categorical variables under a nonparametric prior, which is induced through rounding latent variables having an unknown density with respect to Lebesgue measure. For the proposed class of priors, we provide sufficient conditions for large support, strong consistency and rates of posterior contraction. These conditions allow one to convert sufficient conditions obtained in the setting of multivariate continuous density estimation to the mixed scale case. To illustrate the procedure a rounded multivariate nonparametric mixture of Gaussians is introduced and applied to a crime and communities dataset

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Inconsistency of Pitman-Yor process mixtures for the number of components

In many applications, a finite mixture is a natural model, but it can be difficult to choose an appropriate number of components. To circumvent this choice, investigators are increasingly turning to Dirichlet process mixtures (DPMs), and Pitman-Yor process mixtures (PYMs), more generally. While these models may be well-suited for Bayesian density estimation, many investigators are using them for inferences about the number of components, by considering the posterior on the number of components represented in the observed data. We show that this posterior is not consistent --- that is, on data from a finite mixture, it does not concentrate at the true number of components. This result applies to a large class of nonparametric mixtures, including DPMs and PYMs, over a wide variety of families of component distributions, including essentially all discrete families, as well as continuous exponential families satisfying mild regularity conditions (such as multivariate Gaussians).Comment: This is a general treatment of the problem discussed in our related article, "A simple example of Dirichlet process mixture inconsistency for the number of components", Miller and Harrison (2013) arXiv:1301.270

On approximating copulas by finite mixtures

Copulas are now frequently used to approximate or estimate multivariate distributions because of their ability to take into account the multivariate dependence of the variables while controlling the approximation properties of the marginal densities. Copula based multivariate models can often also be more parsimonious than fitting a flexible multivariate model, such as a mixture of normals model, directly to the data. However, to be effective, it is imperative that the family of copula models considered is sufficiently flexible. Although finite mixtures of copulas have been used to construct flexible families of copulas, their approximation properties are not well understood and we show that natural candidates such as mixtures of elliptical copulas and mixtures of Archimedean copulas cannot approximate a general copula arbitrarily well. Our article develops fundamental tools for approximating a general copula arbitrarily well by a mixture and proposes a family of finite mixtures that can do so. We illustrate empirically on a financial data set that our approach for estimating a copula can be much more parsimonious and results in a better fit than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table