Research Project: Belief and Probability
Probabilistic transformation of belief functions
Affine transformations of belief functions
Epistemic transformations of belief functions
The nexus between belief and probability plays a major role in the theory of evidence, and is the foundation of a popular approach to evidential reasoning called 'transferable belief model'. The problem of finding sensible probabilistic and possibilistic approximations of belief functions has been widely studied by many authors.
As belief functions and probabilities are in the geometric approach simply points of a simplex, the probability transformation problem can be posed and solved in this geometric framework. Each belief function turns out to be associated with two different families of probability transformations, marked by the operator they commute with: affine combination or Dempster's rule.
Starting from the case of binary domains, we identify and study three major geometric entities relating a generic belief function to the set of probabilities P: 1) the dual line connecting belief and plausibility functions; 2) the orthogonal complement of P; and 3) the simplex of consistent probabilities. Each of them is in turn associated with a different probability measure that depends on the original BF. We focus in particular on the geometry and properties of the orthogonal projection of a belief function onto P and its intersection probability, provide their interpretations in terms of degrees of belief, and discuss their behavior with respect to affine combination.
The relative belief of singletons is a recently introduced probability transformation of a belief function. We discussed its nature in terms of degrees of belief under several different angles, its rationale in terms of game theory and its applicability to different classes of belief functions. We proved that it commutes with Dempster's orthogonal sum, and meets a number of properties which are the duals of those met by the relative plausibility of singletons. This highlights a classification of Bayesian approximations in two families, according to the operator they relate to. Finally, we also proposed a natural extension of the relative belief transformation which exists for all belief functions and inherits its properties.
Approximation of credal sets via lower probabilities
Geometry of the epistemic transformations
Credal sets are closed convex sets of probability mass functions. The lower probabilities specified by a credal set for each element of the power set can be used as constraints defining a second credal set. This simple procedure produces an outer approximation, with a bounded number of extreme points, for general credal sets. The approximation is optimal in the sense that no other lower probabilities can specify smaller supersets of the original credal set. Notably, in order to be computed, the approximation does not need the extreme points of the credal set, but only its lower probabilities.
We have seen that the study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form there two homogeneous groups, which we call 'affine' and 'epistemic' families. The epistemic family forms a coherent collection of probability transformations in terms of their behavior with respect to Dempster's rule of combination. We investigated their geometry in both the space of all pseudo belief functions and the probability simplex, and compared it with that of the affine family, providing sufficient conditions under which probabilities of both families coincide.