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Introduction to Dempster-Shafer Theory

Rolf Haenni Center for Junior Research

FellowsUniversity of Konstanz

Contents

- Introductory Example
- Dempster-Shafer Theory
- Conclusion
- Discussion

1. Introductory Example

Evidence 1 Mr. Jones was assassinated by the

Mafia. An informer tells the police that the

selection of the assassin was done as

follows 1) a fair coin is tossed 2) head ?

either Peter or Tom is selected 3) tail ?

either Tom or Mary is selected

Evidence 2 The police finds the assassins

fingerprint. An expert states that it is male

with 80 chance and female with 20 chance.

Using DS belief function

Using probabilities

E2

E1

Tom?

Peter?

Mary?

2. Dempster-Shafer Theory

- Frame of Discernment (set of possible

events, one of them is true) - Multi-Variable Frames
- Belief Potential
- Domain
- Mass Function
- Focal Sets
- Normalized

with

s.t.

(No Transcript)

- Belief is Sub-Additive
- Plausibility is Super-Additive

Additive

Sub-Additive

J. Bernoulli Ars Conjectandi, 1713

?

- Example

7

6

8

1

2

9

3

4

5

H 1,2,3,4,7,8 ?B(H) .4.25 .65 ?P(H)

.4.25.1.05 .8

Hc 5,6,9 ?B(Hc) .2 ?P(Hc) .2.1.05

.35

- Dempsters Rule of Combination

- Non-Monotonicity

- Belief potentials are called
- - vacuous, iff
- - deterministic, iff
- - consonant, iff
- - precise, iff
- - Bayesian, iff

- Causal (conditional) relationships can be

represented by belief potentials - Example

Modeling Knowledge

a) define variables, frames, and domains

b) define independent belief potentials

Quantitative Analysis

Decisions

? according to Smets Transferable Belief Model

(TBM)

- There is a credal level where beliefs are

entertained and a pignistic level where beliefs

are used to make decisions (from pignus a bet

in Latin). - At the credal level beliefs are quantified by

belief functions. - The credal level precedes the pignistic level in

that at any time, beliefs are entertained (and

updated) at the credal level. The pignistic level

appears only when a decision needs to be made. - When a decision must be made, beliefs at the

credal level induce a probability measure at the

pignistic level, i.e. there is a pignistic

transformation from belief functions to

probability functions.

? Insufficient reason principle to define a

probability distribution on n elements, given a

lack of information, give 1/n to each element

Remark

Every probabilistic argumentation system can be

transformed into a set of Dempster-Shafer belief

potentials such that

for

and for all

3. Conclusion

Benefits of Dempster-Shafer Theory

- Allows proper distinction between reasoning and

decision taking - No modeling restrictions (e.g. DAGs)
- It represents properly partial and total

ignorance - Ignorance is quantified
- ? low degree of ignorance means - high

confidence in results - enough information

available for taking decisions - ? high degree of ignorance means - low

confidence in results - gather more information

(if possible) before taking decisions

- Conflict is quantified
- ? low conflict indicates the presence of

confirming information sources ? high conflict

indicates the presence of contradicting sources - Simplicity Dempsters rule of combination covers

- combination of evidence, Bayes rule,

Bayesian updating (conditioning), belief

revision (results from non-monotonicity), etc.

Further Remarks

- Dempster-Shafer theory generalizes probabilistic

(Bayesian) modeling and inference - From a quantitative point of view,

Dempster-Shafer theory is equivalent to

probabilistic argumentation - Dempster-Shafer theory is an instance of Shenoys

valuation-based systems and of Kohlas

information algebras

4. Discussion

DS-Theory is not very successful because

- Inference is less efficient than Bayesian

inference - Pearl is the better speaker than Dempster (and

Shafer, Kohlas, etc.) - Microsoft supports Bayesian Networks
- The UAI community does not like outsiders

DS-Theory is often misunderstood because

- Of most peoples lack of experience in belief

function modeling ? construction of

counter-intuitive examples - Nobody reads Kohlas and Monneys book Theory of

Hints, 1994 - Most people ignore Haenni and Kohlass papers

about Probabilistic Argumentation

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