Bayesian Modeling and Computation for Mixed Data

Multivariate or high-dimensional data with mixed types are ubiquitous in many fields of studies, including science, engineering, social science, finance, health and medicine, and joint analysis of such data entails both statistical models flexible enough to accommodate them and novel methodologies for computationally efficient inference. Such joint analysis is potentially advantageous in many statistical and practical aspects, including shared information, dimensional reduction, efficiency gains, increased power and better control of error rates.

This thesis mainly focuses on two types of mixed data: (i) mixed discrete and continuous outcomes, especially in a dynamic setting; and (ii) multivariate or high dimensional continuous data with potential non-normality, where each dimension may have different degrees of skewness and tail-behaviors. Flexible Bayesian models are developed to jointly model these types of data, with a particular interest in exploring and utilizing the factor models framework. Much emphasis has also been placed on the ability to scale the statistical approaches and computation efficiently up to problems with long mixed time series or increasingly high-dimensional heavy-tailed and skewed data.

To this end, in Chapter 1, we start with reviewing the mixed data challenges. We start developing generalized dynamic factor models for mixed-measurement time series in Chapter 2. The framework allows mixed scale measurements in different time series, with the different measurements having distributions in the exponential family conditional on time-specific dynamic latent factors. Efficient computational algorithms for Bayesian inference are developed that can be easily extended to long time series. Chapter 3 focuses on the problem of jointly modeling of high-dimensional data with potential non-normality, where the mixed skewness and/or tail-behaviors in different dimensions are accurately captured via the proposed heavy-tailed and skewed factor models. Chapter 4 further explores the properties and efficient Bayesian inference for the generalized semiparametric Gaussian variance-mean mixtures family, and introduce it as a potentially useful family for modeling multivariate heavy-tailed and skewed data.