Eigenstructure inference for high-dimensional covariance with generalized shrinkage inverse-Wishart prior

In multivariate statistics, estimating the covariance matrix is essential for understanding the interdependence among variables. In high-dimensional settings, where the number of covariates increases with the sample size, it is well known that the eigenstructure of the sample covariance matrix is inconsistent. The inverse-Wishart prior, a standard choice for covariance estimation in Bayesian inference, also suffers from posterior inconsistency. To address the issue of eigenvalue dispersion in high-dimensional settings, the shrinkage inverse-Wishart (SIW) prior has recently been proposed. Despite its conceptual appeal and empirical success, the asymptotic justification for the SIW prior has remained limited. In this paper, we propose a generalized shrinkage inverse-Wishart (gSIW) prior for high-dimensional covariance modeling. By extending the SIW framework, the gSIW prior accommodates a broader class of prior distributions and facilitates the derivation of theoretical properties under specific parameter choices. In particular, under the spiked covariance assumption, we establish the asymptotic behavior of the posterior distribution for both eigenvalues and eigenvectors by directly evaluating the posterior expectations for two sets of parameter choices. This direct evaluation provides insights into the large-sample behavior of the posterior that cannot be obtained through general posterior asymptotic theorems. Finally, simulation studies illustrate that the proposed prior provides accurate estimation of the eigenstructure, particularly for spiked eigenvalues, achieving narrower credible intervals and higher coverage probabilities compared to existing methods. For spiked eigenvectors, the performance is generally comparable to that of competing approaches, including the sample covariance.