ROBUSTNESS OF BAYES DECISIONS AGAINST THE CHOICE OF PRIOR.

Abstract : The present paper is an attempt at a quantitative evaluation of the approximation that the outcome - in terms of utility or loss - of any Bayes decision procedure may, for large samples, be expected to become insensitive to the choice of prior distribution. More precisely, the author investigates the asymptotic behavior of the difference in loss when using two arbitrary priors, the said difference being evaluated for a fixed 'true' state of nature, consistent with both priors. Under appropriate regularity and identifiability conditions, it turns out that the difference, asymptotically, consists of an error term (a random term with expectation 0) of order n to the -3/2, and a systematic term of order n to the -2. The coefficients depend on the data of the problem, i.e., on the sampling distribution, the loss function, and the priors, in a way indicated, and the subsequent Corollary. He treats only the case of one-dimensional and continuous state and action spaces (for instance, estimation of a single parameter). The case of several parameters seems to offer no essential additional difficulties, provided that the parameter vector (or the relevant component of it) is identifiable under the sampling distribution. The case of discrete decisions (like in testing) offers technical difficulties and may lead to quite different asymptotic properties of the loss. (Author)