Compute the best feasible MSE by estimating the required covariance term in the Bayesian ridge framework

Develop practical techniques to compute the best feasible asymptotic mean squared error specified in equation (F.20) of Theorem F.3 for the high-dimensional linear model y_{t+1} = β' S_t + ε_{t+1} with Gaussian prior β ~ N(0, Σ_β), by estimating the necessary covariance component Σ_β (denoted in the paper as E;) from observed data. The goal is to operationalize the asymptotic expression for the best feasible MSE without access to the true prior covariance and thereby enable empirical implementation of the Bayes-optimal ridge prediction performance bound.

Background

In Appendix F, the authors analyze Bayesian optimality for prediction in the linear model y_{t+1} = β' S_t + ε{t+1} with a Gaussian prior over β. They show the Bayes-optimal predictor coincides with ridge regression under an appropriate mapping and derive an expression for the best feasible asymptotic MSE (equation F.20) that depends on the prior covariance Σβ.

Implementing this bound in practice requires estimating the prior covariance structure, which is nontrivial in high dimensions. The authors explicitly note this estimation challenge and defer it as future research, making the computation of the best feasible MSE a concrete unresolved task.