Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions

Abstract

Skew-normal and skew-t distributions have proved to be useful for capturing skewness and kurtosis in data directly without transformation. Recently, finite mixtures of such distributions have been considered as a more general tool for handling heterogeneous data involving asymmetric behaviors across subpopulations. We consider such mixture models for both univariate as well as multivariate data. This allows robust modeling of high-dimensional multimodal and asymmetric data generated by popular biotechnological platforms such as flow cytometry. We develop Bayesian inference based on data augmentation and Markov chain Monte Carlo (MCMC) sampling. In addition to the latent allocations, data augmentation is based on a stochastic representation of the skew-normal distribution in terms of a random-effects model with truncated normal random effects. For finite mixtures of skew normals, this leads to a Gibbs sampling scheme that draws from standard densities only. This MCMC scheme is extended to mixtures of skew-t distributions based on representing the skew-t distribution as a scale mixture of skew normals. As an important application of our new method, we demonstrate how it provides a new computational framework for automated analysis of high-dimensional flow cytometric data. Using multivariate skew-normal and skew-t mixture models, we could model non-Gaussian cell populations rigorously and directly without transformation or projection to lower dimensions.

Clustering with the multivariate normal inverse Gaussian distribution

“…To address this issue, mixtures of skew-normal or skew-t distributions can be used (Lin et al, 2007b,a;Cabral et al, 2012;Prates et al, 2013a;Vrbik and McNicholas, 2014). However, these distributions can prove numerically unstable in high-dimensional settings (Fruhwirth-Schnatter and Pyne, 2009). …”

Robust mixture of experts modeling using thetdistribution

“…The inference in the previously described approaches is performed by maximum likelihood estimation via expectation-maximization (EM) or extensions (Dempster et al, 1977;McLachlan and Krishnan, 2008), in particular the expectation conditional maximization (ECM) algorithm (Meng and Rubin, 1993). For the Bayesian framework, Frühwirth-Schnatter and Pyne (2010) have considered the Bayesian inference for both the univariate and the multivariate skew-normal and skew-t mixtures. For the regression context, the robust modeling of regression data has been studied namely by Wei (2012) ;Ingrassia et al (2012) who considered a t-mixture model for regression analysis of univariate data, as well as by Bai et al (2012) who relied on the M-estimate in mixture of linear regressions.…”

Risk measures for skew normal mixtures

“…Often data display strong asymmetry, fat tails and multimodality features that are usually shared by different subpopulations. In this context mixtures of asymmetric distributions have been adopted in different areas, see for example Frühwirth-Schnatter and Pyne [6], Bernardi et al [4] and Haas and Mittnik [8]. Among the skewed distrubutions, the Skew Normal of Azzalini [1] and the Skew Student-t of Azzalini and Capitanio [3] have become widely employed because of the attractive properties they share with their symmetric counterparts, and the greater shape flexibility introduced by the additional asymmetry parameter.…”