Optimal information deletion and Bayes' theorem

Arnold Zellner published a seminal paper on Bayes' theorem as an optimal information processing rule, a result that led to the variational formulation of Bayes' theorem, and a central idea in generalized variational inference. Almost 40 years later, we revisit these ideas, but from the perspective of information deletion. We investigate rules that update a posterior distribution into an antedata distribution when a portion of data is removed. In such context, a rule that does not destroy or create nonexistent information is called the optimal information deletion rule and we prove that it coincides with the leave-data-out posterior from Bayes' theorem.