Inference on the Eigenvalues of the Normalized Precision Matrix

Area of Science:

• Statistics
• Machine Learning
• Bayesian Networks

Background:

• Spectral theory of Bayesian Networks requires robust estimation methods for the normalized precision matrix.
• Existing methods for eigenvalue estimation may suffer from bias, particularly under certain data conditions.

Purpose of the Study:

• To derive asymptotic distributions for sample eigenvalues of the normalized precision matrix.
• To develop a second-order bias correction formula for these eigenvalues.
• To propose a Stein-type shrinkage estimator for improved eigenvalue estimation.

Main Methods:

• Derivation of multivariate normal asymptotic distributions for sample eigenvalues.
• Development of a second-order bias correction formula.
• Construction of a Stein-type shrinkage estimator.
• Numerical simulations to compare estimation techniques.

Main Results:

• Asymptotic distributions for sample eigenvalues are provided under general and normal population conditions.
• A formula for second-order bias correction is established.
• A Stein-type shrinkage estimator is proposed.
• Simulations indicate the effectiveness of different methods based on eigenvalue magnitude.

Conclusions:

• The second-order bias-corrected eigenvalue estimator significantly reduces bias when the largest eigenvalue is small.
• For the smallest eigenvalue, the sample eigenvalue or the shrinkage estimator shows less bias.
• The study provides valuable tools for statistical inference in Bayesian Networks.