A correlation-shrinkage prior for Bayesian prediction of the two-dimensional Wishart model

A Bayesian prediction problem for the two-dimensional Wishart model is investigated within the framework of decision theory. The loss function is the Kullback–Leibler divergence. We construct a scale-invariant and permutation-invariant prior distribution that shrinks the correlation coefficient. The prior is the geometric mean of the right invariant prior with respect to permutation of the indices, and is characterized by a uniform distribution for Fisher’s $z$-transformation of the correlation coefficient. The Bayesian predictive density based on the prior is shown to be minimax.