Inference on the Eigenvalues of the Normalized Precision Matrix.

Recent developments in the spectral theory of Bayesian Networks has led to a need for a developed theory of estimation and inference on the eigenvalues of the normalized precision matrix, . In this paper, working under conditions where and remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of under general conditions and under normal populations, a formula for second-order bias correction of these sample eigenvalues, and a Stein-type shrinkage estimator of the eigenvalues. Numerical simulations are performed which demonstrate under what generative conditions each estimation technique is most effective.

Spectral Bayesian network theory.

A Bayesian Network (BN) is a probabilistic model that represents a set of variables using a directed acyclic graph (DAG). Current algorithms for learning BN structures from data focus on estimating the edges of a specific DAG, and often lead to many 'likely' network structures. In this paper, we lay the groundwork for an approach that focuses on learning global properties of the DAG rather than exact edges.