Optimal information deletion and Bayes'theorem

2026-02-08

AI Summary

This study investigates how to reverse-engineer a “counterfactual” data distribution from the posterior that precisely removes specific observed data without introducing or discarding any additional information. We reinterpret Bayes’ theorem as an optimal rule for information deletion, revealing its inherent symmetry and optimality in data removal scenarios. Building upon variational inference and information-theoretic principles, we develop a backward updating framework constrained by strict information conservation—ensuring no creation or destruction of information—and formally demonstrate that this optimal deletion procedure is equivalent to conventional Bayesian updating. This work establishes a theoretical foundation for reversible Bayesian inference and machine unlearning, offering principled insights into how probabilistic models can faithfully forget selected data while preserving overall coherence.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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

Problem

• information deletion
• Bayes' theorem
• posterior distribution
• antedata distribution
• optimal information processing

Innovation

• optimal information deletion
• Bayes' theorem
• antedata distribution
• information processing
• variational inference