Shrinkage priors for circulant correlation structure models
Michiko Okudo, Tomonari Sei
TL;DR
This paper introduces the circulant correlation structure model for multivariate Gaussian data, where $\Sigma = D_\alpha R D_\alpha$ with $R = Q D_\lambda Q^*$ and a scale constraint $\prod_i \lambda_i = 1$. It develops shrinkage priors that damp the non-eigenvalue part of $\Sigma$ via a log-linear $\lambda(\theta)$ parametrization, and proves that Bayesian predictive densities based on these priors asymptotically outperform those based on Jeffreys priors under KL risk. The authors establish optimality results in both a full circulant model and an exchangeable submodel, showing that a uniform shrinkage prior $\pi_S$ is optimal within natural prior families in the high-$\theta$ limit, and that, in the exchangeable case, a family of shrinkage priors yields asymptotic dominance with an explicit optimal setting $\gamma = -1/4$. Numerical illustrations corroborate the theoretical gains, highlighting improved predictive performance due to shrinkage of the non-eigenvalue covariance structure. Overall, the work provides a principled, asymptotically optimal approach to shrinkage in circulant-correlation models with practical implications for multivariate prediction under structured covariance.
Abstract
We consider a new statistical model called the circulant correlation structure model, which is a multivariate Gaussian model with unknown covariance matrix and has a scale-invariance property. We construct shrinkage priors for the circulant correlation structure models and show that Bayesian predictive densities based on those priors asymptotically dominate Bayesian predictive densities based on Jeffreys priors under the Kullback-Leibler (KL) risk function. While shrinkage of eigenvalues of covariance matrices of Gaussian models has been successful, the proposed priors shrink a non-eigenvalue part of covariance matrices.