Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models

The problem of estimating a covariance matrix in small samples has been con sidered by several authors following early work by Stein. This problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects. We propose a set of hierarchical pr iors (HPs) for the covariance matrix that produce posterior shrinkage towar d a specified structure-here we examine shrinkage reward diagonality. We th en address the computational difficulties raised by incorporating these pri ors, and nonconjugate priors in general, into hierarchical models. We apply a combination of approximation, Gibbs sampling (possibly with a Metropolis step), and importance reweighting to fit the models, and compare this hybr id approach to alternative Markov Chain Monte Carlo methods. Our investigat ion involves three alternative HPs. The first works with the spectral decom position of the covariance matrix and produces both shrinkage of the eigenv alues toward each other and shrinkage of the rotation matrix toward the ide ntity. The second produces shrinkage of the correlations toward 0, and the third uses a conjugate Wishart distribution to shrink toward diagonality. A simulation study shows that the first two HPs can be very effective in red ucing small-sample risk, whereas the conjugate Wishart version sometimes pe rforms very poorly. We evaluate the computational algorithm in the context of a normal nonlinear random-effects model and illustrate the methodology w ith a logistic random-effects model.