This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The models are then applied in the context of financial time series for predictive portfolio analysis providing a significant improvement in performance of optimal investment decisions.

This paper presents a default model-selection procedure for Gaussian graphical models that involves two new developments. First, we develop an objective version of the hyper-inverse Wishart prior for restricted covariance matrices, called the HIW g-prior, and show how it corresponds to the implied fractional prior for covariance selection using fractional Bayes factors. Second, we apply a class of priors that automatically handles the problem of multiple hypothesis testing implied by covariance selection. Numerical experiments show that these priors strongly control the number of false edges included in the model, thereby automatically rewarding sparsity. We demonstrate our methods on a variety of simulated examples, concluding with a real example analyzing covariation in mutual-fund returns. These studies reveal that the combined use of a multiplicity-correction prior on graphs with the hyper-inverse Wishart g-prior on covariance matrices yields better performance than conventional covariance selection methods.

We propose a new stochastic search algorithm for Gaussian graphical models called the mode oriented stochastic search. Our algorithm relies on the existence of a method to accurately and efficiently approximate the marginal likelihood associated with a graphical model when it cannot be computed in closed form. To this end, we develop a new Laplace approximation method to the normalizing constant of a G-Wishart distribution. We show that combining the mode oriented stochastic search with our marginal likelihood estimation method leads to excellent results with respect to other techniques discussed in the literature. We also describe how to perform inference through Bayesian model averaging based on the reduced set of graphical models identified. Finally, we give a novel stochastic search technique for multivariate regression models.

We develop Bayesian analysis of matrix-variate normal data with conditional independence graphical structuring of the characterising variance matrix parameters. This leads to fully Bayesian analysis of matrix normal graphical models, including discussion of novel prior specifications, the resulting problems of posterior computation addressed using Markov chain Monte Carlo methods, and graphical model assessment that involves approximate evaluation of marginal likelihood functions under specified graphical models. Modelling and inference for spatial/image data via a novel class of Markov random fields that arise as natural examples of matrix normal graphical models is discussed. This is complemented by the development of a broad class of dynamic models for matrix-variate time series within which stochastic elements defining time series errors and structural changes over time are subject to graphical model structuring. Three examples illustrate these developments and highlight questions of graphical model uncertainty and comparison in matrix data contexts.

Abstract: A parametrization of hypergraphs based on the geometry of points in R d is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or Markov structure of multivariate distributions. Specifically, we can recover both the junction tree factorization as well as the hyper Markov law. This approach offers greater control on the distribution of graph features than Erdös-Rényi random graphs, supports inference of factorizations that cannot be retrieved by a graph alone, and leads to new Metropolis/Hastings Markov chain Monte Carlo algorithms with both local and global moves in graph space. We illustrate the utility of this parametrization and prior specification using simulations.

We describe a comprehensive framework for performing Bayesian inference for Gaussian graphical models based on the G-Wishart prior with a special focus on efficiently including nondecomposable graphs in the model space. We develop a new approximation method to the normalizing constant of a G-Wishart distribution based on the Laplace approximation. We review recent developments in stochastic search algorithms and propose a new method, the mode oriented stochastic search (MOSS), that extends these techniques and proves superior at quickly finding graphical models with high posterior probability. We then develop a novel stochastic search technique for multivariate regression models and conclude with a real-world example from the recent covariance estimation literature. Supplemental materials are available online.

Markov distributions are used to describe multivariate data with conditional independence structure. Applications of Markov distributions arise in many fields including demography, flood prediction, and telecommunications. A hyper Markov law is a distribution over the space of all Markov distributions; such laws have been used as prior distributions for various types of graphical models. Dirichlet processes have also been used to specify priors in a non-parametric form. I have developed a family of non-parametric hyper Markov laws that I call hyper Dirichlet processes, which combine the separate ideas of hyper Markov laws and non-parametric prior processes. In my thesis, I propose to describe these distributions and their properties, and to apply them to specific problems. For example, I define a hyper Markov mixture of Gaussians and use it in the form of a hyper Markov prior to provide a non-parametric way to mix graphical Gaussian distributions. 1

We introduce and explore a new class of stationary time series models for variance matrices based on a constructive definition exploiting inverse Wishart distribution theory. The main class of models explored is a novel class of stationary, first-order autoregressive (AR) processes on the cone of positive semi-definite matrices. Aspects of the theory and structure of these new models for multivariate “volatility ” processes are described in detail and exemplified. We then develop approaches to model fitting via Bayesian simulation-based computations, creating a custom filtering method that relies on an efficient innovations sampler. An example is then provided in analysis of a multivariate electroencephalogram (EEG) time series in neurological studies. We conclude by discussing potential further developments of higherorder AR models and a number of connections with prior approaches. 1. Introduction. Modeling

variable and covariance selection with an