A New Approach to Updating Beliefs (1304.1119v1)

Abstract: We define a new notion of conditional belief, which plays the same role for Dempster-Shafer belief functions as conditional probability does for probability functions. Our definition is different from the standard definition given by Dempster, and avoids many of the well-known problems of that definition. Just as the conditional probability Pr (lB) is a probability function which is the result of conditioning on B being true, so too our conditional belief function Bel (lB) is a belief function which is the result of conditioning on B being true. We define the conditional belief as the lower envelope (that is, the inf) of a family of conditional probability functions, and provide a closed form expression for it. An alternate way of understanding our definition of conditional belief is provided by considering ideas from an earlier paper [Fagin and Halpern, 1989], where we connect belief functions with inner measures. In particular, we show here how to extend the definition of conditional probability to non measurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given our lack of information. Our definition of conditional belief turns out to be an exact analogue of our definition of inner conditional probability.

• The paper introduces a conditional belief function defined as the infimum of conditional probabilities, addressing limitations of the standard DS method.
• It offers a closed-form expression for conditional belief and plausibility functions that adheres to the sure thing principle for improved consistency.
• The study highlights challenges like non-commutativity in belief updates, stressing trade-offs between computational ease and informational precision.

A New Approach to Updating Beliefs: Overview and Critical Analysis

The paper authored by Ronald Fagin and Joseph Y. Halpern presents an innovative perspective on updating beliefs in the context of Dempster-Shafer (DS) belief functions. Recognizing the limitations and counterintuitive results produced by the standard DS approach, Fagin and Halpern propose a new definition of conditional belief. This new formulation seeks to align more closely with the principles of probability theory while avoiding the shortcomings associated with DS conditioning.

Key Contributions of the Paper

The paper introduces a novel operational definition for conditional belief that mirrors the function of conditional probability in traditional probability spaces. The authors redefine conditional belief functions as the lower envelope of a family of conditional probability functions, providing a closed-form expression that ensures consistent results across various applications. Key components of their new approach include:

1. Conditional Belief Function: The paper defines Bel(·|B), a conditional belief for event B, as the infimum of all conditional probabilities consistent with the original belief function. This is a departure from the DS definition, which frequently yields results that appear paradoxical or inconsistent, particularly in the case of non-measurable sets.
2. Closed-form Expression: The paper provides an elegant closed-form solution for these conditional belief and plausibility functions, facilitating easier computation and application.
3. The Sure Thing Principle: The authors demonstrate that their new definition adheres to the sure thing principle, thus aligning the behavior of conditional beliefs closer to probabilistic intuition.
4. Lack of Commutativity in Belief Updates: Unlike traditional probability functions, which are commutative under sequential updates, the new belief updating method may not exhibit commutativity, highlighting a unique challenge in the belief function paradigm.

Discussion and Implications

The proposed definition attempts to bridge the gap between traditional probability theory and the flexibility offered by belief functions in handling uncertainty. Notably, when belief functions are conveyed as generalized probabilities, maintaining the probabilistic characteristics during belief updates can be critical for their consistent application in domains like expert systems and AI-driven decision-making processes.

The approach also highlights the potential information loss when abstracting a set of probability functions into belief and plausibility bounds. This trade-off between computational feasibility and informational richeness would need careful consideration depending on the application's complexity and accuracy requirements. Furthermore, the inability to guarantee that updates commute when using belief functions might impact systems where order of information receipt is crucial.

This paper's implications could lead to more robust frameworks for uncertainty management in AI, particularly when integrating diverse evidence sources. Future research can stem from exploring the reconciliation of DS rule of combination with these new conditional belief definitions or investigating further conditions under which belief function updates could potentially commute.

Conclusion

Through this new perspective, Fagin and Halpern contribute to the refinement of the DS theory of evidence, situating belief functions within a framework that aspires for consistency with established probabilistic conventions without the perplexing outcomes associated with traditional DS updates. Their work opens discussions for expanding the use of belief functions in fields necessitating rigorous handling of uncertainty, and sets a foundation for subsequent studies aiming to enhance the efficacy of belief function applications.