Expectation consistency of inverse probability distributions

Abstract

SUMMARY Inverse probability distributions for inference about a discrete parameter with an infinity of possible values are considered, based on discrete data. Their expectation consistency with the conditional probability distributions of the data is defined and shown to be implied if the inverse distributions are generalized Bayes posterior distributions. It is shown that, if each data point has nonzero probability for every parameter value, then Bayesianity is also necessary. A counterexample to the removal of the condition of nonzero probability is presented.