Belief Revision

When revising beliefs, make the minimal necessary change to accommodate the new evidence. Preserve as many of the old beliefs as possible. This principle, often called Informational Economy, prevents unnecessary and arbitrary belief fluctuations, ensuring stability in the agent's world model.

• Formal Goal: Minimize the set difference between the old belief set K and the new set K*A.

The revised belief set should be deductively closed. If an agent believes P and P → Q, it should also believe Q. This ensures the agent's beliefs are logically integrated and that implicit consequences of its explicit beliefs are recognized, maintaining a complete and rational model.

Formalized by the AGM postulates, this principle states that if you retract a belief A and then later learn A again, you should recover all original beliefs that depended on A. Symbolically: K ⊆ (K*A) + A. This ensures revision is not overly destructive and allows for belief reinstatement in a principled way.

The outcome of belief revision should depend on the logical content of the evidence, not its syntactic form. Learning "P & Q" should lead to the same revised state as learning "Q & P". This principle is critical for iterated revision, where an agent undergoes multiple, sequential belief updates, ensuring the process is robust and deterministic.

A computational technique for maintaining a probability distribution over a set of competing belief states (hypotheses) as sequential, noisy evidence is observed. It's a practical implementation of probabilistic belief revision over time.

• Core Algorithm: Often implemented via extensions to the Kalman filter (for continuous states) or multiple hypothesis tracking algorithms that manage a branching tree of possibilities.
• Key Challenge: Combats cognitive commitment by explicitly keeping unlikely explanations alive, preventing premature convergence on a single flawed belief state.
• Use Case: Critical in target tracking (radar/sonar), diagnostic systems where multiple faults are possible, and natural language understanding to maintain multiple interpretations of ambiguous dialogue.

A specific type of non-monotonic inference that applies typical-case assumptions in the absence of specific information to the contrary. It represents a common-sense 'jump to conclusion' that is later subject to belief revision.

• Formal Rule: 'Birds typically fly.' From 'Tweety is a bird,' we defeasibly conclude 'Tweety flies.' Upon learning 'Tweety is a penguin,' this conclusion is retracted.
• Role in Revision: Default conclusions create the provisional beliefs that form the substrate for future revision operations when exceptional evidence is encountered.
• Computational Frameworks: Implemented in Default Logic (Raymond Reiter) and is a core behavior of logic programming languages like Prolog with negation-as-failure.

The normative, probabilistic framework for revising degrees of belief (credences) in light of new evidence, governed by Bayes' theorem. It is the quantitative counterpart to logical belief revision.

• Core Formula: P(H|E) = [P(E|H) * P(H)] / P(E). It precisely calculates the posterior probability of a hypothesis H given evidence E.
• Contrast with Logical Revision: While logical revision deals with binary true/false beliefs, Bayesian updating handles graded beliefs (probabilities). The principle of conservatism is enforced by using the prior P(H).
• Application: The foundation for probabilistic graphical models, Bayesian networks, and optimal sensor fusion, providing a mathematically rigorous method for belief revision under uncertainty.

The process of integrating multiple, potentially conflicting belief sets from different sources (e.g., multiple agents, sensors, or experts) into a single, consistent collective belief set. It generalizes belief revision to multi-source input.

• Key Distinction: Belief revision handles a single agent's beliefs vs. new evidence. Belief merging handles multiple prior belief sets with equal or weighted standing.
• Operators: Uses merging operators (like Δ) that take multiple belief bases (K1, K2,... Kn) and output a merged base Δ(K1,..., Kn), satisfying postulates for fairness, consistency, and majority representation.
• Use Case: Essential for multi-agent system coordination, database integration, and jury-style deliberation in ensemble AI systems.