[Paper Review] Belief Revision with Uncertain Inputs in the Possibilistic Setting

Didier Dubois, Henri Prade|arXiv (Cornell University)|Feb 13, 2013

Logic, Reasoning, and Knowledge24 references3 citations

TL;DR

This paper proposes a belief revision framework under uncertain inputs using possibility theory, comparing min-based and product-based conditioning operations. It integrates Williams' transmutations and Boutilier's natural revision within the possibilistic setting, showing how uncertain evidence can be consistently incorporated via ordinal or numerical possibility scales.

ABSTRACT

This paper discusses belief revision under uncertain inputs in the framework of possibility theory. Revision can be based on two possible definitions of the conditioning operation, one based on min operator which requires a purely ordinal scale only, and another based on product, for which a richer structure is needed, and which is a particular case of Dempster's rule of conditioning. Besides, revision under uncertain inputs can be understood in two different ways depending on whether the input is viewed, or not, as a constraint to enforce. Moreover, it is shown that M.A. Williams' transmutations, originally defined in the setting of Spohn's functions, can be captured in this framework, as well as Boutilier's natural revision.

Motivation & Objective

• To address belief revision when incoming information is uncertain, extending classical revision models to handle imprecise or partial knowledge.
• To compare two conditioning operations—min-based (ordinal) and product-based (cardinal)—for revising possibility distributions.
• To unify existing revision approaches like Williams' transmutations and Boutilier's natural revision within the possibilistic framework.
• To clarify whether uncertain inputs should be treated as constraints to enforce or as mere sources of information.
• To provide a principled way to revise beliefs using possibility theory that accommodates both qualitative and quantitative uncertainty.

Proposed method

• Uses possibility theory to represent degrees of belief, with possibility distributions over possible worlds.
• Applies two conditioning operations: min-based (requiring only ordinal scale) and product-based (needing numerical scales, equivalent to Dempster's rule).
• Defines belief revision as updating a possibility distribution using a conditional possibility distribution derived from uncertain input.
• Introduces a distinction between treating uncertain inputs as constraints to be enforced or as information to be integrated.
• Reformulates Williams' transmutations and Boutilier's natural revision as specific instances of possibilistic revision under different conditioning rules.
• Employs a preference-based semantics where higher possibility values indicate more plausible worlds, and revision adjusts these values consistently.

Experimental results

Research questions

• RQ1 How can belief revision be formalized when inputs are uncertain, rather than certain?
• RQ2 What are the differences and trade-offs between min-based and product-based conditioning in possibility theory?
• RQ3 Can Williams' transmutations be captured within the possibilistic framework, and if so, how?
• RQ4 How does Boutilier's natural revision relate to possibilistic conditioning operations?
• RQ5 Should uncertain inputs be treated as constraints to be enforced, or as information to be integrated in a flexible way?

Key findings

• The min-based conditioning operation allows revision using only ordinal scales, making it suitable for qualitative belief revision.
• The product-based conditioning operation requires numerical possibility values and corresponds to a special case of Dempster's rule of conditioning.
• Williams' transmutations are naturally captured in the possibilistic setting through specific revision rules based on possibility distributions.
• Boutilier's natural revision is shown to be equivalent to a particular form of possibilistic revision using the product-based conditioning rule.
• The paper establishes that treating uncertain inputs as constraints leads to more rigid revision, while treating them as information allows for more flexible belief updates.
• The framework provides a unified perspective on various revision strategies, showing their compatibility within possibility theory.