Dempster-Shafer Theory - PowerPoint Presentation

Slide1

Dempster-Shafer Theory

SIU CS 537

4/12/11 and 4/14/11

Chet LanginSlide2

Dempster, A. P. (1967). "Upper and

Lower

Probabilities Induced

by

a Multivalued Mapping

.“

The

Annals of Mathematical Statistics

38

(2): 325-339

.

Shafer, G. (1976).

A Mathematical Theory of Evidence

,

Princeton

University Press

.Slide3

What is Dempster-Shafer?

Dempster-Shafer (D-S or DS)

Mathematical theory of evidence.

Data fusion. Degree of belief.

Generalization of Bayes theory.Sets. Mass, not probability.“Bel Function” – Belief functionSlide4

The D-S Environment

(Theta):

The elements are all mutually exclusive.

All of the possible elements in the universe are in the set and so the set is exhaustive.

Each subset of

can be interpreted as a possible answer to a question.

Since the elements are mutually exclusive and the environment is exhaustive, there can be only one correct answer to a question.

 Slide5

D-S Environment, Cont.

(Theta):

All the possible subset of

Fig. 5.6, Page 281 (Airliner, Bomber, Fighter).

An environment is called a

Frame of Discernment

where its elements may be interpreted as possible answers, and only one answer is correct.

 Slide6

D-S Environment, Cont.

(Theta):

A set of size

N

has exactly

subsets, including itself, and these subsets define the Power Set (

):

The Power Set of the environment has as its elements all answers to the possible questions of the Frame of Discernment.

 Slide7

D-S vs. Probability

In D-S Theory, the

Degree of Belief

in evidence is analogous to the mass of a physical object (mass of evidence supports a belief). Evidence measure

amount of the mass

Basic Probability Assignment (BPA).

Fundamental difference between D-S Theory and probability theory is the treatment of ignorance.

Principle of indifference

 Slide8

Non-belief vs. Ignorance

D-S does not force belief to be assigned to ignorance. Instead, the mass is assigned only to those subsets of the environment to which you wish to assign belief.

Not assigned belief

no belief or non-belief. Should be associated with the environment

. Disbelief

non-belief.

.

Every set in the Power Set of the environment which has a mass greater than zero is a Focal Element.

 Slide9

D-S Mass

Mass is a function that maps each element of the Power Set into a real number in the

interval.

By conversion:

 Slide10

Combining Evidence

First radar data:

Second radar data:

 Slide11

D-S Rule of Combination

Extends over all elements whose intersection

.

denotes the orthogonal sum or direct sum which is defined by summing the mass product intersections on the right-hand side of the rule.

The new mass is a consensus because it tends to favor agreement rather than disagreement.

 Slide12

Example Combination

Bomber = 0.63 + 0.27 = 0.90

Bomber or Fighter = 0.07

Non-belief = 0.03

 Slide13

Range of Belief

represents the belief of a bomber, only, but

and

imply additional information since their sets include a bomber

It is

plausible

that their orthogonal sums may contribute to a belief in the bomber.

Plausible that it might be a bomber.

 Slide14

What Would Make Plausibility < 1?

(Airliner)

 Slide15

Evidential Interval

The true

Range of Belief

is somewhere in the range of 0.9 to 1.0. Also called the

Evidential Interval. The lower bound (0.9) is called Support (spt) in evidential reasoning. It is called

Bel

in D-S Theory. The upper bound (1.0) is called

Plausibility (pls). In general:(See Table 5.5 in text) Slide16

Example Evidential Intervals

[1, 1] Completely True

[0, 0] Completely False

[0, 1] Completely Ignorant

[Bel, 1] Tends to support[0, Pls] Tends to refute[Bel,

Pls

] Tends to both support & refuteSlide17

Bel vs. Bel

()

Bel

is belief, a part of the evidence interval. It refers to one set.

Bel() is a function that is the total belief of a set and all its subsets.

Bel

function

belief measure Slide18

Bel() Example

All the mass that supports a set. Is more global.

 Slide19

Combination of 2 Bel()

The combination of 2 belief functions (as in mass) can be expressed in terms of orthogonal sums of the masses of a set and all its subsets:

 Slide20

The Normalization of Belief

Suppose a third sensor is provided:

 Slide21

A New Table

 Slide22

Normalization

Divide each element by

1-k

where

k is defined for any set X and

Y

as:

 Slide23

Normalization, Cont.

OK!Slide24

New Evidential Interval

Belief | Plausibility |

Disbelief

Belief | Plausibility