Dempster-Shafer Theory
SIU CS 537
4/12/11 and 4/14/11
Chet Langin
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.
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 function
The D-S Environment
•
•
•
•
Θ (Theta): Θ = {𝜃1 , 𝜃2 , 𝜃3 , … , 𝜃𝑛 }
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 Θ 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.
D-S Environment, Cont.
Θ (Theta): Θ = {𝜃1 , 𝜃2 , 𝜃3 , … , 𝜃𝑛 }
• All the possible subset of 𝜃 = {𝐴, 𝐵, 𝐹} → 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.
D-S Environment, Cont.
Θ (Theta): Θ = {𝐴, 𝐵, 𝐹}
• A set of size N has exactly 2𝑁 subsets,
including itself, and these subsets define the
Power Set (𝑃(Θ)):
𝑃(Θ) = {∅, 𝐴 , 𝐵 , 𝐹 , 𝐴, 𝐵 , 𝐴, 𝐹 , 𝐵, 𝐹 , 𝐴, 𝐵, 𝐹 }
• The Power Set of the environment has as its
elements all answers to the possible questions
of the Frame of Discernment.
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 ≡ 𝑚 ≡ Basic Probability
Assignment (BPA).
• Fundamental difference between D-S Theory and
probability theory is the treatment of ignorance.
𝑃=
1
𝑁
↔ Principle of indifference
𝑃 𝐻 + 𝑃 𝐻′ = 1
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 Θ. Disbelief ≢
non-belief. 𝑚1 𝐵, 𝐹 = 0.7. 𝑚1 Θ = 0.3.
• Every set in the Power Set of the environment which
has a mass greater than zero is a Focal Element.
D-S Mass
• Mass is a function that maps each element of
the Power Set into a real number in the [0,1]
interval.
𝑚: 𝑃 Θ → [0,1]
By conversion: 𝑚 ∅ = 0
𝑚 𝑋 =1
∀𝑋,𝑋∈𝑃(Θ)
Combining Evidence
First radar data:
𝑚1 𝐵, 𝐹 = 0.7
Second radar data:
𝑚2 𝐵 = 0.9
𝑚1 Θ = 0.3
𝑚2 Θ = 0.1
D-S Rule of Combination
𝑚1 ⨁𝑚2 𝑍 =
𝑚1 (𝑋)𝑚2 (𝑌)
𝑋∩𝑌=𝑍
Extends over all elements whose intersection
𝑋 ∩ 𝑌 = 𝑍.
⨁ 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.
Example Combination
𝑚2 𝐵
𝑚1 𝐵, 𝐹
= 0.7
𝑚1 Θ = 0.3
𝑚3 ⟶
= 0.9
𝑚2 Θ = 0.1
𝐵 0.63
𝐵, 𝐹 0.07
𝐵 0.27
Θ 0.03
Bomber = 0.63 + 0.27 = 0.90
Bomber or Fighter = 0.07
Non-belief = 0.03
Range of Belief
𝑚3 ({𝐵}) represents the belief of a bomber, only,
but 𝑚3 ({𝐵, 𝐹}) and 𝑚3 (Θ) imply additional
information since their sets include a bomber
⟹
It is plausible that their orthogonal sums may
contribute to a belief in the bomber.
0.9 + 0.07 + 0.03 = 1.0
⟹
Plausible that it might be a bomber.
What Would Make Plausibility < 1?
𝑚3 𝐴 = 0.1
(Airliner)
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 DS Theory. The upper bound (1.0) is called
Plausibility (pls). In general:
0 ≤ Bel ≤ Pls ≤ 1
(See Table 5.5 in text)
Example Evidential Intervals
[1, 1]
[0, 0]
[0, 1]
[Bel, 1]
[0, Pls]
[Bel, Pls]
Completely True
Completely False
Completely Ignorant
Tends to support
Tends to refute
Tends to both support & refute
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
Bel() Example
Bel 𝑋 =
𝑚(𝑌)
𝑌⊆𝑋
Bel1 𝐵, 𝐹
= 𝑚1 𝐵, 𝐹
+ 𝑚1 𝐵
+ 𝑚1 ({𝐹})
0.7 + 0 + 0 = 0.7
All the mass that supports a set. Is more global.
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:
𝐵𝑒𝑙1 ⨁𝐵𝑒𝑙2 𝐵
𝐵𝑒𝑙1 ⨁𝐵𝑒𝑙2 𝐵, 𝐹
= 𝑚1 ⨁𝑚2 𝐵 + 𝑚1 ⨁𝑚2 ∅
= 0.90 + 0 = 0.90
= 𝑚1 ⨁𝑚2 𝐵, 𝐹 + 𝑚1 ⨁𝑚2 𝐹
= 0.07 + 0 + 0.90 = 0.97
+ 𝑚1 ⨁𝑚2 𝐵
The Normalization of Belief
Suppose a third sensor is provided:
𝑚3 𝐴
= 0.95
𝑚3 𝜃
= 0.05
A New Table
𝑚3 𝐴
= 0.95
𝑚3 Θ = 0.05
𝒎𝟏 ⨁𝒎𝟐 ({𝑩})
𝟎. 𝟗𝟎
𝒎𝟏 ⨁𝒎𝟐 ({𝑩, 𝑭})
𝟎. 𝟎𝟕
𝒎𝟏 ⨁𝒎𝟐 (𝚯)
𝟎. 𝟎𝟑
∅0
∅0
𝐴 0.0285
𝐵 0.045
𝐵, 𝐹 0.0035
Θ 0.0015
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐴
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐵
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐵, 𝐹
𝑚1 ⨁𝑚2 ⨁𝑚3 Θ
𝑚1 ⨁𝑚2 ⨁𝑚3 ∅
= 0.0285
= 0.0450
= 0.0035
= 0.0015
= 0.0000
= 0.0785
≠ 1 ‼!
Normalization
Divide each element by 1-k where k is defined
for any set X and Y as:
𝑘=
𝑚1 (𝑋)𝑚2 (𝑌)
𝑋∩𝑌=∅
𝑘 = 0.95 × 0.90 + 0.95 × 0.07
= 0.8550 + 0.0665 = 0.9215
Normalization, Cont.
1 − 𝑘 = 1 − 0.9215 = 0.0785
0.0285
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐴 =
= 0.363
0.0785
0.0450
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐵 =
= 0.573
0.0785
0.0035
𝑚1 ⨁𝑚2 ⨁𝑚3 𝐵, 𝐹 =
= 0.045
0.0785
0.0015
𝑚1 ⨁𝑚2 ⨁𝑚3 𝜃 =
= 0.019
0.0785
0.0000
𝑚1 ⨁𝑚2 ⨁𝑚3 ∅ =
= 0.000
0.0785
= 1.000
OK!
New Evidential Interval
Belief | Plausibility
Belief | Plausibility | Disbelief