Title: Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices,
Abstract: We obtain the optimal Bayesian minimax rate for the unconstrained large
covariance matrix of multivariate normal sample with mean zero, when both the
sample size, n, and the dimension, p, of the covariance matrix tend to
infinity. Traditionally the posterior convergence rate is used to compare the
frequentist asymptotic performance of priors, but defining the optimality with
it is elusive. We propose a new decision theoretic framework for prior
selection and define Bayesian minimax rate. Under the proposed framework, we
obtain the optimal Bayesian minimax rate for the spectral norm for all rates of
p. We also considered Frobenius norm, Bregman divergence and squared
log-determinant loss and obtain the optimal Bayesian minimax rate under certain
rate conditions on p. A simulation study is conducted to support the
theoretical results.