1. Background
Red be observed • Bel: Bel({1})=0, Bel({1, 4})=1 • Pl: Pl({1})=1, Pl({1, 4})=1
Special report on evidence theory
Xiaolu Ke
11/57
Authorities with their contribution
1. Background
19/47
Special report on evidence theory
Xiaolu Ke
12/57
Authorities with their contribution
1. Background
Arthur P. Dempster • ‘Upper and lower probabilities induced by a multivalued mapping’ [1] —— Dempster’s rule of combination • Expectation Maximum (EM) algorithm Glenn Shafer • < A mathematical theory of evidence theory > [2]
m
Bel
Pl
Q
Mobius transform
Special report on evidence theory Xiaolu Ke 10/57
Bel and Pl of Dice example No observation • Bel: Bel({1})=0, Bel({1, 2, 3, 4, 5, 6})=1 • Pl: Pl({1})=1, Pl({1, 2, 3, 4, 5, 6})=1
P ( ) 0 P ( ) 1 P ( A A ) P ( A ) P ( A ), A A 1 2 1 2 1 2
Plausibility function (upper probability)
Pl ( A)
A B
m( B )
Xiaolu Ke
15/57
Authorities with their contribution
1. Background
Xi’an Jiaotong University
• Han Deqiang, Yang Yi Northwestern Polytechnical University • Pan Quan, Liu Zhunga Fuzhou University • Wang Yingming
Special report on evidence theory
X
Background
• • • Two simple examples Basic concepts Authorities with their contribution
Theoretic research directions
{1 , , n }
• Constraint conditions
m( ) 0; m( A) 0, A ;
A
m( A) 1.
(1)
• Explanation of m
m({1 , 2 }) 1 m({1}) m({ 2 })
Special report on evidence theory Xiaolu Ke 8/57
Special report on evidence theory
Xiaolu Ke
5/57
Dice throw example
1. Background
No observation • P(1)=1/6 If red is observed • P(1)=1/2
Principle of Indifference!
• Theoretic research directions
• • • Evidence conflict measure Combination rules Approximation of belief functions
• Applications
• • Possible applications Evidential KNN
m( A) a, m( B) b, A B
m( A) a, m( B ) b, A B , | A | 1, | B | 1
m1 (1 ) 0.9, m1 ( 2 ) 0.1 m1 (1 ) 0.45, m1 ( 2 ) 0.55 VS m2 (1 ) 0.55, m2 ( 2 ) 0.45 m2 (1 ) 0.55, m2 ( 2 ) 0.45
Special report on evidence theory
Xiaolu Ke
6/57
Represent exactly what we know
1. Background
The Ace Example • One ace: m({ (A, A), (A, 2), (A, 2), (A, 2), (A, 2)} )=1 • One ace of heart: m({ (A, A), (A, 2), (A, 2)} )=1 • One ace of spade: m({ (A, A), (A, 2), (A, 2)} )=1 The Dice Example • No observation: m({1, 2, 3, 4, 5, 6})=1 • Red be observed: m({1, 4})=1
• Background
• • • Two simple examples Basic concepts Authorities with their contribution
• Theoretic research directions
• • • Evidence conflict measure Combination rules Approximation of belief functions
Special report on evidence theory
Xiaolu Ke
7/57
Basic Concepts
1. Background
Evidence • Helpful information provided by an independent source Basic probability assignment (BPA) • Universe (frame of discernment)
Xiaolu Ke
4/57
The Ace example
1. Background
Condition 1: One ace { (A, A), (A, 2), (A, 2), (A, 2), (A, 2)} => P((A, A))=1/5 Condition 2: One ace of heart { (A, A), (A, 2), (A, 2)} => P((A, A))= 1/3 Condition 3: One ace of spade { (A, A), (A, 2), (A, 2)} => P((A, A))= 1/3
• • • Evidence conflict measure Combination rules Approximation of belief functions
Applications
• • Possible applications Evidential KNN
Special report on evidence theory
Special Report
An Introduction to Evidence Theory
Lab of vibration control and vehicle control, USTC
2014/12/9
Special report on evidence theory 1/57
Outline
Related terms • Distance/Dissimilarity/Disagreement
Special report on evidence theory
Xiaolu Ke
18/57
Typical methods Conflict coefficient (Shafer) • Definition
Jean Dezert & Florentin Smarandache
• PCR1 – PCR5 [5]
Special report on evidence theory
Xiaolu Ke
14/57
Authorities with their contribution
1. Background
Xiaolu Ke
17/57
Evidence conflict measure
2. Theoretic research directions
Purposes • Choose proper combination rules • Evaluate reliability of sources • Approximate belief functions Difficulties • Uncertainty • Non-specificity • Accordance
Thierry Denœux • Applications to pattern recognition: EKNN, ECM, etc • Cautious rule [6] Fabio Cuzzolin • Geometric approach [7,8]