Given a sufficient statistic, basic predictive inference based on frequency theory actually implies the existence of a prediction distribution function, conditional on the sufficient statistic. Unlike Bayesian posterior predictive functions, the derived distribution is not necessarily a valid one. If it is, the prediction distribution function is necessarily a Bayesian predictive function. In suchcases, the frequency theory prediction method implies a particular Bayesian prior on the nuisance parameter, thus these prediction methods represent a special case of Bayesian predictive inference.
