Dempster-Shafer theory

# Dempster-Shafer theory

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The Dempster–Shafer theory (DST) is a mathematical theory of evidence
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...

. It allows one to combine evidence from different sources and arrive at a degree of belief (represented by a belief function) that takes into account all the available evidence. The theory was first developed by Arthur P. Dempster
Arthur P. Dempster
Arthur Pentland Dempster is a Professor Emeritus in the Harvard University Department of Statistics. He was one of four faculty when the department was founded in 1957.He was a Putnam Fellow in 1951. He obtained his Ph.D. from Princeton University in 1956...

and Glenn Shafer.

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.

## Overview

Dempster–Shafer theory is a generalization of the Bayesian theory of subjective probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

; 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 that contradict those arrived at using probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

.

Often used as a method of sensor fusion
Sensor fusion
Sensor fusion is the combining of sensory data or data derived from sensory data from disparate sources such that the resulting information is in some sense better than would be possible when these sources were used individually...

, 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 distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

. 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 non-empty 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 — so that — then the set of all subsets — the power set — is called . 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 (2|S| in general), ranging from the empty subset (0, 0, 0, 0, 0) to the “everything” subset (1, 1, 1, 1, 1). The empty subset represents a contradiction, which is not true in any state, and is thus assigned a mass of zero; the remaining masses are normalised so that their total is 1. The “everything” subset is often labelled “unknown” as it represents the state where all elements are present, in the sense that you cannot tell which is actual.

### Belief and plausibility

Shafer's framework allows for belief about propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility:
beliefplausibility.

Belief in a hypothesis is constituted by the sum of the masses of all sets enclosed by it (i.e. the sum of the masses of all subsets of the hypothesis). It is the amount of belief that directly supports a given hypothesis at least in part, forming a lower bound. Plausibility is 1 minus the sum of the masses of all sets whose intersection with the hypothesis is 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.

For example, suppose we have a belief of 0.5 and a plausibility of 0.8 for a proposition, say “the cat in the box is dead.” This means that we have evidence that allows us to state strongly that the proposition is true with a confidence of 0.5. However, the evidence contrary to that hypothesis (i.e. “the cat is alive”) only has a confidence of 0.2. The remaining mass of 0.3 (the gap between the 0.5 supporting evidence on the one hand, and the 0.2 contrary evidence on the other) is “indeterminate,” meaning that the cat could either be dead or alive.
This interval represents the level of uncertainty based on the evidence in your system.
Hypothesis Mass Belief Plausibility
Null (neither alive nor dead) 0 0 0
Alive 0.2 0.2 0.5
Dead 0.5 0.5 0.8
Either (alive or dead) 0.3 1.0 1.0

The null hypothesis is set to zero by definition (it corresponds to “no solution”). The orthogonal hypotheses “Alive” and “Dead” have probabilities of 0.2 and 0.5, respectively. This could correspond to “Live/Dead Cat Detector” signals, which have respective reliabilities of 0.2 and 0.5. Finally, the all-encompassing “Either” hypothesis (which simply acknowledges there is a cat in the box) picks up the slack so that the sum of the masses is 1. The belief for the “Alive” and “Dead” hypotheses matches their corresponding masses because they have no subsets; belief for “Either” consists of the sum of all three masses (Either, Alive, and Dead) because “Alive” and “Dead” are each subsets of “Either”. The “Alive” plausibility is 1 − m (Dead) and the “Dead” plausibility is 1 − m (Alive). Finally, the “Either” plausibility sums m(Alive) + m(Dead) + m(Either). The universal hypothesis (“Either”) will always have 100% belief and plausibility —it acts as a checksum
Checksum
A checksum or hash sum is a fixed-size datum computed from an arbitrary block of digital data for the purpose of detecting accidental errors that may have been introduced during its transmission or storage. The integrity of the data can be checked at any later time by recomputing the checksum and...

of sorts.

Here is a somewhat more elaborate example where the behavior of belief and plausibility begins to emerge. We're looking through a variety of detector systems at a single faraway signal light, which can only be coloured in one of three colours (red, yellow, or green):
Hypothesis Mass Belief Plausibility
Null 0 0 0
Red 0.35 0.35 0.56
Yellow 0.25 0.25 0.45
Green 0.15 0.15 0.34
Red or Yellow 0.06 0.66 0.85
Red or Green 0.05 0.55 0.75
Yellow or Green 0.04 0.44 0.65
Any 0.1 1.0 1.0

Events of this kind would not be modeled as disjoint sets in probability space as they are here in mass assignment space. Rather the event "Red or Yellow" would be considered as the union of the events "Red" and "Yellow", and (see the axioms of probability theory) P(Red or Yellow) ≥ P(Yellow), and P(Any)=1, where Any refers to Red or Yellow or Green. In DST the mass assigned to Any refers to the proportion of evidence that can't be assigned to any of the other states, which here means evidence that says there is a light but doesn't say anything about what color it is. In this example, the proportion of evidence that shows the light is either Red or Green is given a mass of 0.05. Such evidence might, for example, be obtained from a R/G color blind person. DST lets us extract the value of this sensor's evidence. Also, in DST the Null set is considered to have zero mass, meaning here that the signal light system exists and we are examining its possible states, not speculating as to whether it exists at all.

### Combining beliefs

Beliefs corresponding to independent pieces of information are combined using Dempster's rule of combination, which is a generalization of the special case of Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....

where events are independent. Note that the probability masses from propositions that contradict each other can also be used to obtain a measure of how much conflict there is in a system. This measure has been used as a criterion for clustering multiple pieces of seemingly conflicting evidence around competing hypotheses.

In addition, one of the computational advantages of the Dempster–Shafer framework is that priors and conditionals need not be specified, unlike Bayesian methods, which often use a symmetry (minimax error) argument to assign prior probabilities to random variables (e.g. assigning 0.5 to binary values for which no information is available about which is more likely). However, any information contained in the missing priors and conditionals is not used in the Dempster–Shafer framework unless it can be obtained indirectly—and arguably is then available for calculation using Bayes equations.

Dempster–Shafer theory allows one to specify a degree of ignorance in this situation instead of being forced to supply prior probabilities that add to unity. This sort of situation, and whether there is a real distinction between risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...

and ignorance
Ignorance
Ignorance is a state of being uninformed . The word ignorant is an adjective describing a person in the state of being unaware and is often used as an insult...

, has been extensively discussed by statisticians and economists. See, for example, the contrasting views of Daniel Ellsberg, Howard Raiffa
Howard Raiffa
Howard Raiffa is the Frank P. Ramsey Professor of Managerial Economics, a joint chair held by the Business School and the Kennedy School of Government at Harvard University...

, Kenneth Arrow and Frank Knight
Knightian uncertainty
In economics, Knightian uncertainty is risk that is immeasurable, not possible to calculate.Knightian uncertainty is named after University of Chicago economist Frank Knight , who distinguished risk and uncertainty in his work Risk, Uncertainty, and Profit:- Common-cause and special-cause :The...

.

## Formal definition

Let X be the universal set
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a set of all sets leads to a paradox...

: the set representing all possible states of a system under consideration. The power set

is the set of all subsets of X, including the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

. For example, if:

then

The elements of the power set can be taken to represent propositions concerning the actual state of the system, by containing all and only the states in which the proposition is true.

The theory of evidence assigns a belief mass to each element of the power set. Formally, a function

is called a basic belief assignment (BBA), when it has two properties. 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, a given member of the power set, 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 have, by definition, their 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 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:

And conversely, for finite A, given the belief measure bel(B) for all subsets B of A, we can find the masses m(A) with the following inverse function:

where |A − B| is the difference of the cardinalities of the two sets.

It follows from the last two equations that, for a finite set X, you need know only 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. In the case of an infinite X, there can be well-defined belief and plausibility functions but no well-defined mass function.

## Dempster's rule of combination

The problem we now face is how to combine two independent sets of mass assignments. That is, how do we combine evidence from difference sources? We do this through Dempster's rule of combination. 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 m1 and m2 in the following manner:

where

K is a measure of the amount of conflict between the two mass sets.

### Effects of conflict

The normalization factor above, 1 − K, has the effect of completely ignoring conflict and attributing any mass associated with conflict to the null set. This combination rule for evidence can therefore produce counterintuitive results when there is significant conflict, as we show next.

#### Example with low conflict

The following example, where Dempster's rule is more appropriate, results from reversing the probability values of the preceding example.
Suppose that one doctor believes a patient has either a brain tumor— with a probability of 0.99—or meningitis—with a probability of only 0.01. A second doctor also believes the patient has a brain tumor—with a probability of 0.99—and believes the patient suffers from concussion—with a probability of only 0.01. If we calculate m (brain tumor) with Dempster’s rule, we obtain

This result implies complete support for the diagnosis of a brain tumour, which both doctors believed very likely. The agreement arises from the low degree of conflict between the two sets of evidence comprised by the two doctors' opinions.

In either case, it would be reasonable to expect that:

since the existence of non-zero belief probabilities for other diagnoses implies less than complete support for the brain tumour diagnosis.

#### Example with high conflict

The following example has been introduced by Zadeh in 1979 , ,
to point out the counter-intuitive result generated by Dempster's rule.

Suppose that one has two equireliable doctors and one doctor believes a patient has either a brain tumor— with a probability (i.e. a basic belief assignment - bba's, or mass of belief) of 0.99—or meningitis—with a probability of only 0.01. A second doctor believes the patient has a concussion —with a probability of 0.99—and believes the patient suffers from meningitis—with a probability of only 0.01. Applying Dempster’s rule to combine these two sets of masses of belief, one gets finally m(meningitis)=1 (the meningitis is diagnosed with 100 percent of confidence). Such result goes against the common sense since both doctors agree that there is a little chance that the patient has a meningitis.

This very interesting example has been the starting point of many research works for trying to find a solid justification for Dempster's rule and for foundations of Dempster-Shafer Theory , , or to show the inconsistencies of this theory , , .

## Criticism

Judea Pearl
Judea Pearl
Judea Pearl is a computer scientist and philosopher, best known for developing the probabilistic approach to artificial intelligence and the development of Bayesian networks ....

(1988a, chapter 9; 1988b and 1990); has argued that it is misleading to interpret belief functions as representing either “probabilities of an event,” or “the confidence one has in the probabilities assigned to various outcomes,” or “degrees of belief (or confidence, or trust) in a proposition,” or “degree of ignorance in a situation.” Instead, belief
functions represent the probability that a given proposition is provable from a set of other propositions, to which probabilities are assigned. Confusing probabilities of truth with probabilities of provability may lead to counterintuitive results in reasoning tasks such as (1) representing incomplete knowledge, (2) belief-updating and (3) evidence pooling. He further demonstrated that, if partial knowledge is encoded and updated by belief function methods, the resulting beliefs cannot serve as a basis for rational decisions.

Kłopotek and Wierzchoń

proposed to interpret the Dempster–Shafer theory in terms of statistics of decision tables (of the rough set theory), whereby the operator of combining evidence should be seen as relational join of decision tables. In another interpretation

M.A. Kłopotek and S.T. Wierzchoń
propose to view this theory as describing destructive material processing (under loss of properties), e.g. like in some semiconductor production processes. Under both interpretations reasoning in DST gives correct results, contrary to the earlier probabilistic interpretations, criticized by Pearl in the cited papers and by other researchers.

## See also

• Imprecise probability
Imprecise probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify...

• Upper and lower probabilities
Upper and lower probabilities
Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.Because...

• Possibility theory
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...

• Probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

• Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....

• Bayesian network
Bayesian network
A Bayesian network, Bayes network, belief network or directed acyclic graphical model is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph . For example, a Bayesian network could represent the probabilistic...

• G. L. S. Shackle
G. L. S. Shackle
George Lennox Sharman Shackle was an English economist. He made a practical attempt to challenge classical rational choice theory and has been characterised as a "post-Keynesian," though he is influenced as well by Austrian economics; he has been described as drawing "Keynesian conclusions from...

• Transferable belief model
Transferable belief model
The transferable belief model is an elaboration on the Dempster-Shafer theory of evidence.-Context:Consider the following classical problem of information fusion. A patient has an illness that can be caused by three different factors A, B and C...

• Info-gap decision theory
Info-gap decision theory
Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the...

• Subjective logic
Subjective logic
Subjective logic is a type of probabilistic logic that explicitly takes uncertainty and belief ownership into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and incomplete knowledge...

• Doxastic logic
• Linear belief function
Linear Belief Function
Linear Belief Function is an extension of the Dempster-Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally...

## Further reading

• Yager, R. R., & Liu, L. (2008). Classic works of the Dempster–Shafer theory of belief functions. Studies in fuzziness and soft computing, v. 219. Berlin: Springer
Springer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...

. ISBN 9783540253815.
• more references