The Dempster-Shafer theory
is a mathematical theory of evidence
based on belief functions
and plausible reasoning
, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. The theory was developed by Arthur P. Dempster
and Glenn Shafer
Consider two possible gambles
The first gamble is that we bet on a head turning up when we toss a coin that is known to be fair. Now consider the second gamble, in which we bet on the outcome of a fight between the world's greatest boxer and the world's greatest wrestler. Assume we are fairly ignorant about martial arts and would have great difficulty making a choice of who to bet on.
Many people would feel more unsure about taking the second gamble, in which the probabilities are unknown, rather than the first gamble, in which the probabilities are easily seen to be one half for each outcome. Dempster-Shafer theory allows one to consider the confidence one has in the probabilities assigned to the various outcomes.
be the universal set
: the set of all states under consideration. The power set
, is the set of all possible sub-sets of X
, including the empty set
. For example, if:
The elements of the power set can be taken to represent propositions that one might be interested in, by containing all and only the states in which this proposition is true.
The theory of evidence assigns a belief mass to each subset of the power set. Formally, a function , is called a basic belief assignment (BBA), when it verifies two axioms. First, the mass of the empty set is zero:
Second, the masses of the remaining members of the power set add up to a total of 1:
The mass m(A) of a given member of the power set, A, expresses the proportion of all relevant and available evidence that supports the claim that the actual state belongs to A but to no particular subset of A. The value of m(A) pertains only to the set A and makes no additional claims about any subsets of A, each of which has, by definition, its own mass.
From the mass assignments, the upper and lower bounds of a probability interval can be defined. This interval contains the precise probability of a set of interest (in the classical sense), and is bounded by two non-additive continuous measures called belief (or support) and plausibility:
The belief bel(A) for a set A is defined as the sum of all the masses of (not necessarily proper) subsets of the set of interest:
The plausibility pl(A) is the sum of all the masses of the sets B that intersect the set of interest A:
The two measures are related to each other as follows:
It follows from the above that you need know but one of the three (mass, belief, or plausibility) to deduce the other two, though you may need to know the values for many sets in order to calculate one of the other values for a particular set.
Dempster's rule of combination
The problem we now face is how to combine two independent sets of mass assignments. The original combination rule, known as Dempster's rule of combination, is a generalization of Bayes' rule
. This rule strongly emphasises the agreement between multiple sources and ignores all
the conflicting evidence through a normalization factor. Use of that rule has come under serious criticism when significant conflict in the information is encountered.
Specifically, the combination (called the joint mass) is calculated from the two sets of masses and in the following manner:
is a measure of the amount of conflict between the two mass sets. The normalization factor, , has the effect of completely ignoring conflict and attributing any mass associated with conflict to the null set. Consequently, this operation yields counterintuitive results in the face of significant conflict in certain contexts.
Dempster-Shafer theory is a generalization of the Bayesian theory of subjective probability; whereas the latter requires probabilities for each question of interest, belief functions base degrees of belief (or confidence, or trust) for one question on the probabilities for a related question. These degrees of belief 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 which 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 entities that comprise a system. Suppose for example that a system has five members, that is to say five independent states, exactly one of which is actual. If the original set is called S, , then the set of all subsets —the power set— is called 2S. Since you can express each possible subset as a binary vector (describing whether any particular member is present or not by writing a “1” or a “0” for that member's slot), it can be seen that there are 25 subsets possible (