The Transferable Belief Model

Abstract

We describe the transferable belief model, a model for representing quantified beliefs based on belief functions. Beliefs can be held at two levels: (1) a credal level where beliefs are entertained and quantified by belief functions, (2) a pignistic level where beliefs can be used to make decisions and are quantified by probability functions. The relation between the belief function and the probability function when decisions must be made is derived and justified. Four paradigms are analyzed in order to compare Bayesian, upper and lower probability, and the transferable belief approaches.

A Contingency Analysis of LeActiveMath’s Learner Model

“…It combines a simple issue-based approach [9], in which issues related to content items are identified in their metadata, with a generic multidimensional framework for learner models and belief functions as numeric knowledge representations [10,11]. The mechanisms involved in the learning modelling process, from interpreting input information to deriving the corresponding evidence and finally updating beliefs based on it are sketched in figure 1.…”

“…Table 2 contains details of information and decisions that can be inferred from the mass distributions shown in table 1. These are pignistic distributions, which are probability distributions derived from mass distributions [11], single value summaries 4 and final decisions on the actual learner levels that would result from beliefs justified only by the single pieces of evidence in table 1. The table shows that XLM cannot make decisions under complete ignorance, yet it can be forced to make a decision in very close cases, as in the third and fourth rows in the table.…”

New Semantics for Quantitative Possibility Theory

“…Unfortunately, it occurs that You only know the value of the 'pignistic' probabilities the source would use to bet on the actual value of Ω (Smets, 1990;Smets & Kennes, 1994). The pignistic probability induced by a belief function is built by defining a uniform probability on each set of positive mass, and performing the convex mixture of these probabilities according to the mass function.…”

“…In the objective context, quantitative possibility can be devised as an approximation of upper and lower frequentist probabilities, due to the presence of incomplete statistical observations (Dubois & Prade, 1986a;Gebhardt & Kruse, 1993). In the subjective context, quantitative possibility theory somehow competes with the probabilistic model in its personalistic or Bayesian views and with the transferable belief model (TBM) (Smets & Kennes, 1994;Smets, 1997Smets, , 1998, both of which also intend to represent degrees of belief. A major issue when developing formal models that represent psychological quantities (belief is such an object) is to produce an operational definition of what these degrees are supposed to quantify.…”

“…They claim that any state of incomplete knowledge of an agent can be modeled by a single probability distribution on the appropriate referential, and that the probabilities can be revealed by a betting experiment in which the agent provides betting odds under an exchangeable bet assumption. A similar setting exists for imprecise probabilities (Walley, 1991), relaxing the assumption of exchangeable bets, and more recently for the TBM as well (Smets & Kennes, 1994;Smets, 2001), introducing several betting frames corresponding to various partitions of the referential. In that sense, the numerical values encountered in these three models are well defined.…”