Credal semantics of bayesian transformations in terms of probability intervals

Area of Science:

• Decision Theory
• Probability Theory
• Artificial Intelligence

Background:

• Belief functions (b.f.) represent uncertainty.
• Classical Bayesian transformations are widely used.
• Existing methods lack a unified geometric interpretation for interval probabilities derived from b.f.s.

Purpose of the Study:

• To propose a credal representation for interval probabilities associated with belief functions.
• To establish connections between this representation and classical Bayesian transformations.
• To provide a geometric framework for understanding these transformations.

Main Methods:

• Geometric representation of interval probabilities using pairs of upper and lower simplices.
• Utilizing the notion of 'focus' for pairs of simplices.
• Interpreting the pignistic function as the center of mass of the credal set.

Main Results:

• The credal representation of interval probability is geometrically defined by pairs of simplices.
• Relative belief of singletons, relative plausibility of singletons, and intersection probability are identified as foci of specific simplex pairs.
• Demonstrated a link between belief functions, interval probabilities, and Bayesian transformations via geometric foci.

Conclusions:

• The proposed credal representation offers a unified geometric perspective on Bayesian transformations of belief functions.
• This framework facilitates the formulation of models analogous to the transferable belief model.
• Provides a novel approach to understanding uncertainty representation and transformation in AI and decision theory.