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    This study develops methods for estimating eigenvalues of the normalized precision matrix in Bayesian Networks. Bias-corrected and shrinkage estimators improve accuracy, especially for extreme eigenvalues.

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    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.