© 2002 by Biometrika Trust
Asymptotic approximations to posterior distributions via conditional moment equations
1 Western Ecological Research Center, U.S.Geological Survey, Sacramento, California 95826, U.S.Ajulie.yee{at}usgs.gov 2 Department of Statistics, University of California, Davis, California 95616, U.S.A. johnson{at}wald.ucdavis.edu samanieg{at}wald.ucdavis.edu
We consider asymptotic approximations to joint posterior distributions in situations where the full conditional distributions referred to in Gibbs sampling are asymptotically normal.Our development focuses on problems where data augmentation facilitates simpler calculations, but results hold more generally. Asymptotic mean vectors are obtained as simultaneous solutions to fixed point equations that arise naturally in the development. Asymptotic covariance matrices flow naturally from the work of Arnold & Press (1989) and involve the conditional asymptotic covariance matrices and first derivative matrices for conditional mean functions. When the fixed point equations admit an analytical solution, explicit formulae are subsequently obtained for the covariance structure of the joint limiting distribution, which may shed light on the use of the given statistical model. Two illustrations are given.
Key Words: Bayesian approach; Data augmentation; EM algorithm; Fixed point theorem; Gibbs sampling; Latent data; Screening data
Received September 2000. Revised January 2002