Prediction Sufficiency When the Loss Function Does Not Depend on The Unknown Parameter

Abstract

It is shown by Takeuchi and Akahira, 1974, that conditional independence together with a condition of "partial sufficiency" imply "prediction sufficiency" for loss functions not depending on the unknown parameter. We shall here prove that these conditions are necessary as well and thereby obtain a complete description, in terms of conditional expectations, of "prediction sufficiency" for loss functions not depending on the unknown parameter. It turns out that these conditions may be replaced by a condition of conditional independence for prior distributions.