Implementing high dimensional Gaussian models for financial applications
Keane, Kevin R.
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We discuss a number of issues surrounding the modeling of high dimensional Gaussian data in a non-stationary environment. The combination of high dimension and few relevant observations due to evolution of the environment demands that learning be efficient and well regularized. The two frameworks in which we achieve efficient learning and regularization are Bayesian dynamic linear mixture models and maximum entropy models. The original theoretical contributions of this dissertation include a method for carrying forward prior distributions in Bayesian dynamic linear models as the environment's eigenspace evolves; a clarification of a pervasive and longstanding misinterpretation of the algorithm demonstrated in the seminal Gaussian graphical model (GGM) paper "Covariance Selection"; a new parametrization for multivariate Gaussian sample statistics facilitating transparent understanding of the precision matrix and variance matrix obtained with a maximum entropy algorithm; and, a critical commentary on the hazards of using GGMs for inference.