A representation of the posterior mean for a location model

Abstract

SUMMARY An exact representation of the posterior mean is developed for a location model and the class of priors that are normal scale mixtures. The result makes use of the conditional distribution of the maximum likelihood estimator and Masreliez's theorem. Directions for future development are indicated. An exact representation for the posterior mean, E(01y), is given where y is a 1 x n vector of observations from a location model, f(x - 0), and 0 has a prior density, p(0), that is a normal scale mixture. Let L(0) denote the likelihood function and let y = (0, a) where 0 is the maximum likelihood estimator and a is the maximal ancillary. The representation makes use of two results: the conditional distribution of the maximum likelihood estimator, p( 00, a) (Barndorff-Nielsen, 1983), and a result of Masreliez (1975). It is shown that, under a normal prior, E(Ojy) can be represented as a linear transformation of the score function of p (01 a), where p (0la)= p(10, a)p(0) dO. The representation can be viewed as a generalization of Masreliez's result that deals with the model, X = 0 + , 0 - N(m, r2) and represents the posterior mean in terms of the derivative of the logarithm of the marginal density for X. The Appendix gives a sufficient regularity condition and a rigorous proof of an interchange of derivative and integral sign required for Masreliez's theorem. A representation is also developed in the case where the prior density is a normal scale mixture. The results have a quantitative robustness appeal. For example, under any model and a normal prior the sensitivity of E( Oy) with respect to aberrant observations is