On the generalization of the likelihood function and the likelihood principle

The parametric likelihood and likelihood principle (LP) play a central role in parametric methodology and in the foundations of statistics.The main purpose of this article is to extend the concepts of likelihood and LP to general inferential aims and models covering, for example, prediction and empirical Bayes models. The likelihood function is the joint distribution of the data and the unobserved variables of inferential interest, considered as a function of the parameters and these inferential variables. LP is based on this likelihood, and the principles of sufficiency (SP) and conditionality (CP) are modified such that the equivalence SP and CP double left right arrow LP is valid, generalizing Birnbaum's theorem.