Dempster–Shafer theory

The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty—a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called belief function) that takes into account all the available evidence. The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty—a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called belief function) that takes into account all the available evidence. In a narrow sense, the term Dempster–Shafer theory refers to the original conception of the theory by Dempster and Shafer. However, it is more common to use the term in the wider sense of the same general approach, as adapted to specific kinds of situations. In particular, many authors have proposed different rules for combining evidence, often with a view to handling conflicts in evidence better. The early contributions have also been the starting points of many important developments, including the transferable belief model and the theory of hints. Dempster–Shafer theory is a generalization of the Bayesian theory of subjective probability. Belief functions base degrees of belief (or confidence, or trust) for one question on the probabilities for a related question. The degrees of belief themselves may or may not have the mathematical properties of probabilities; how much they differ depends on how closely the two questions are related. Put another way, it is a way of representing epistemic plausibilities but it can yield answers that contradict those arrived at using probability theory. Often used as a method of sensor fusion, Dempster–Shafer theory is based on two ideas: obtaining degrees of belief for one question from subjective probabilities for a related question, and Dempster's rule for combining such degrees of belief when they are based on independent items of evidence. In essence, the degree of belief in a proposition depends primarily upon the number of answers (to the related questions) containing the proposition, and the subjective probability of each answer. Also contributing are the rules of combination that reflect general assumptions about the data. In this formalism a degree of belief (also referred to as a mass) is represented as a belief function rather than a Bayesian probability distribution. Probability values are assigned to sets of possibilities rather than single events: their appeal rests on the fact they naturally encode evidence in favor of propositions. Dempster–Shafer theory assigns its masses to all of the subsets of the propositions that compose a system—in set-theoretic terms, the power set of the propositions. For instance, assume a situation where there are two related questions, or propositions, in a system. In this system, any belief function assigns mass to the first proposition, the second, both or neither. Shafer's formalism starts from a set of possibilities under consideration, for instance numerical values of a variable, or pairs of linguistic variables like 'date and place of origin of a relic' (asking whether it is antique or a recent fake). A hypothesis is represented by a subset of this frame of discernment, like '(Ming dynasty, China)', or '(19th century, Germany)'.:p.35f. Shafer's framework allows for belief about such propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility: In a first step, subjective probabilities (masses) are assigned to all subsets of the frame; usually, only a restricted number of sets will have non-zero mass (focal elements).:39f. Belief in a hypothesis is constituted by the sum of the masses of all subsets of the hypothesis-set. It is the amount of belief that directly supports either the given hypothesis or a more specific one, thus forming a lower bound on its probability. Belief (usually denoted Bel) measures the strength of the evidence in favor of a proposition p. It ranges from 0 (indicating no evidence) to 1 (denoting certainty). Plausibility is 1 minus the sum of the masses of all sets whose intersection with the hypothesis is empty. Or, it can be obtained as the sum of the masses of all sets whose intersection with the hypothesis is not empty. It is an upper bound on the possibility that the hypothesis could be true, i.e. it “could possibly be the true state of the system” up to that value, because there is only so much evidence that contradicts that hypothesis. Plausibility (denoted by Pl) is defined to be Pl(p) = 1 − Bel(~p). It also ranges from 0 to 1 and measures the extent to which evidence in favor of ~p leaves room for belief in p.