Coherent information deletion: Bayes' theorem and generalized Bayesian unlearning

Abstract

Bayes' theorem admits an information-processing interpretation due to Zellner (1988): under the Shannon-information criterion, the posterior is the unique rule that processes prior and data information without information loss. We revisit these ideas, but from the perspective of information deletion. Given a posterior based on a complete dataset, what distribution should replace it when a subset of the data is removed? We define information deletion using the same information conservation principle as Zellner (1988), and show that the optimalpost-deletion distribution is exactly the leave-data-out posterior. We then extend the framework beyond likelihood-based inference from Bayes to the generalized Bayesian updating of Bissiri et al. (2016) based on loss functions. We introduce a sequential coherence requirement for deletion, under which, removing two pieces of information jointly is equivalent to removing them successively. The resulting coherent deletion rule exactly recovers the generalized Bayesian posterior based only on the retained data. Restricting these optimization problems to variational families yields corresponding formulations of variational Bayesian and generalized Bayesian unlearning.