Introduction to Dempster-Shafer Theory

1. AI Application Please enter the unknown probability: _ Introduction to Dempster-Shafer Theory Rolf Haenni Center for Junior Research FellowsUniversity of Konstanz Contents: Introductory Example Dempster-Shafer Theory Conclusion Discussion

2. 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 assassin’s fingerprint. An expert states that it is male with 80% chance and female with 20% chance.

3. 0.8 Tom? Peter Tom Peter? 0.2 Mary0.2 Mary 0.5 0.5 E1 E2 Peter Tom Mary p2(Peter) = p2(Tom) = 0.4p2(Mary) = 0.2 p1(Peter) = p1(Mary) = 0.25p1(Tom) = 0.5 ? ? Using DS belief function: Using probabilities: E2 E1 Tom? Peter? Mary?

4. 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.

5. Normalized Mass Function: • Belief Function: • Plausibility Function: • Properties:  •  •  • 

6. Y H E X Z H Y E X • Belief is Sub-Additive: • Plausibility is Super-Additive: Additive: Sub-Additive: J. Bernoulli: “Ars Conjectandi”, 1713

7. []m({6}) = .2 []m({1,2,8}) = .4 []m({2,4,5,9}) = .1 []m({3,4}) = .25 []m({1,2,3,4,5,6,7,8,9}) = .05 • Ignorance of  relative to H: • Total ignorance:  • 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

8. Marginalization: • Dempster’s Rule of Combination: • Non-Monotonicity:

9. xy xy xy xy xy x¬y x¬y x¬y x¬y x¬y X Y ¬xy ¬xy ¬xy ¬xy ¬xy ¬x¬y ¬x¬y ¬x¬y ¬x¬y ¬x¬y • Belief potentials are called • - vacuous, iff • - deterministic, iff • - consonant, iff • - precise, iff • - Bayesian, iff • Causal (conditional) relationships can be represented by belief potentials • Example: = =

10. with 5  Knowledge Base  Hypergraph H E 3 D A B C 9 F K 2 G J M H 4 1 L 8 7 a) define query domain Q and query b) compute  local computation c) derive belief and plausibility Modeling Knowledge: a) define variables, frames, and domains b) define independent belief potentials: Quantitative Analysis:

11. CREDAL LEVEL Bel Pl 0 1 BetP 0 1 PIGNISTIC LEVEL  Pignistic transformation: Decisions:  according to Smets’ “Transferable Belief Model” (TBM) • There is a credal levelwhere beliefs are entertained and a pignistic levelwhere 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 transformationfrom 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

12. and Degree ofSupport Arguments Belief Probabilistic ArgumentationSystem Dempster-Shafer BeliefPotentials Hypothesis Hypothesis Degree ofPossibility Counter-Arguments Plausibility Remark: Every probabilistic argumentation system can be transformed into a set of Dempster-Shafer belief potentials such that for and for all

13. 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

14. Conflict is quantified: •  low conflict indicates the presence of confirming information sources  high conflict indicates the presence of contradicting sources • Simplicity: Dempster’s 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 Shenoy’s valuation-based systems and of Kohlas’ information algebras

15. 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 people‘s lack of experience in belief function modeling  construction of „counter-intuitive“ examples • Nobody reads Kohlas‘ and Monney‘s book „Theory of Hints“, 1994 • Most people ignore Haenni and Kohlas‘s papers about „Probabilistic Argumentation“