Characterizations of Prediction Sufficiency (Adequacy) in Terms of Risk Functions

Abstract

Prediction sufficiency (adequacy), as it is usually defined in terms of conditional expectations, does imply "real" prediction sufficiency; i.e. sufficiency in terms of risk functions. The converse holds provided we permit the loss to depend on the unknown parameter. This is no longer true if we insist on loss functions which do not involve the unknown parameter. Conditional independence still holds but ordinary sufficiency may fail. If, however, we require equivalence of risk functions, then ordinary sufficiency and, consequently, prediction sufficiency follows.