Bayes convolution

A general convolution theorem within a Bayesian framework is presented. Con sider estimation of the Euclidean parameter theta by an estimator T within a parametric model. Let W be a prior distribution for theta and define G as the W-average of the distribution of T - theta under theta. In some cases, for any estimator T the distribution G can be written as a convolution G = K star L with K a distribution depending only on the model, i.e, on W and the distributions under theta of the observations. In such a Bayes convolut ion result optimal estimators exist, satisfying G = K, For location models we show that finite sample Bayes convolution results hold in the normal, lo ggamma and exponential case, Under regularity conditions we prove that norm al and loggamma are the only smooth location cases. We also discuss relatio ns with classical convolution theorems.