证据理论(杨建波教授学术稿)

– It is a process of drawing a conclusion from evidence
– The conclusion is supposed to be probable or uncertain – The conclusion is updated with accumulation of evidence – There is a need to combine multiple pieces of evidence
Θ {h1 ,, hN }
Evidence represented as ei mi hn , pi (hn ) , n 1, , N probability distribution:
Orthogonal s百度文库m of two m1 m2 (ht ) pieces of evidence:
• Limitations of current probabilistic reasoning
• Evidential reasoning rule for evidence combination
• Illustrative examples of evidential reasoning • Applications in multiple criteria decision analysis
• Applications in multiple criteria decision analysis
9
Generalised Probabilistic Reasoning
Disease Diagnosis Example – Case III
Given:
Expert 1: "I am 80% sure it's meningitis, but 90% sure it is NOT brain tumor or concussion.”
(The joint conclusion of the two experts)
5
Inference with Uncertain Evidence
Disease Diagnosis Example – Case II
Given:
Expert 1: "I am 50% sure it's meningitis, but there is a chance of 50% that it's concussion.” Expert 2: "I am 50% sure it's a brain tumor, but there is a chance of 50% that it's concussion.”
Orthogonal sum operation:
m m1 m2
0, p B ,1 pC , 2 [m1 m2 ]( ) B C , 1 pB ,1 pC , 2 B C
This is the degree of belief that two pieces of evidence are in complete conflict, i.e. with B∩C=Ø This is the degree of belief that both pieces of evidence point to hypothesis 13 θ somehow, i.e. with B∩C=θ
11
Generalised Probabilistic Reasoning
Basic concepts and evidence representation
Power set of
N : the 2 subsets of Θ h1, , hN Θ


, h ,, h , N 1 Θ P(Θ) 2 h1 , h2 ,, h1 , hN , {h1 ,, hN 1}, Θ
Infer: “It is probable that the patient has meningitis,
concussion and/or brain tumor.”
Question: “What are the probabilities of the patient having
meningitis, concussion and/or brain tumor?”
Question: “What are the probabilities of the patient having
meningitis, concussion and/or brain tumor?”
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Generalised Probabilistic Reasoning
Disease Diagnosis Example – Case III
Disease Diagnosis Example – Case II:
Degree of belief Meningitis Concussion Tumor
Expert 1 (m1)
Expert 2 (m2)
m1 m2
Before normalisation
0.5
0 0
0.5
0.5 0.25
{C, T}
{M,C,T}
Expert 1
Expert 2 Expert 3
0.8
0.4 0.1
0
0.3 0.3
0
0 0.5
0.1
0.2 0
0.1
0 0
0
0.1 0
0
0 0.1
Given 90% sure it is NOT brain tumor and 80% sure it is meningitis, there should be 10% belief assigned to meningitis or concussion {M, C}
6
Bayesian Inference – Conjunctive Reasoning
Bayes’ or Dempster’s rule to combine independent evidence
Frame of discernment:
mutually exclusive & collectively exhaustive hypotheses

p1 (ht ) p2 (ht )
N n 1
p1 (hn ) p2 (hn )
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Dempster’s rule represented in the form equivalent to Bayes’ rule
Dempster’s Rule – Conjunctive Reasoning
To combine two pieces of independent evidence
From Bayesian Inference to Evidential Reasoning
For Decision Making under Uncertainty
Professor Jian-Bo Yang (杨剑波)
Professor Dong-Ling Xu (徐冬玲) Decision and Cognitive Sciences Research Centre (www.mbs.ac.uk/DSRC) Manchester Business School The University of Manchester, UK
Question 1: “How to represent these experts’ judgments?”
Assignment of basic probabilities to sets of hypotheses
Degree of Belief

0
0 0
{M}
{C}
{T}
{M, C} {M, T}
4
Inference with Certain Evidence
Disease Diagnosis Example – Case I Given: Expert 1: "I am sure it's concussion.” Expert 2: "I am sure it's concussion.” Infer: “The patient has concussion.”
Evidence represented e m i i as belief distribution:
, p , Θ
,i
12
Generalised Probabilistic Reasoning
Dempster’s rule for evidence combination
3
Bayesian Inference
Example – Disease Diagnosis
Context: A doctor reasons about possible diseases of a patient, which have been narrowed down to three possibilities: meningitis (M), concussion (C) and/or brain tumor (T). The doctor consults two medical experts, who independently give him their judgments. The question is how to infer a conclusion about the disease of the patient from experts’ judgments
Main Topics of the Presentation
• An introduction to Bayesian inference • Generalised probabilistic reasoning and Dempster’s rule • Limitations of current probabilistic reasoning • Evidential reasoning rule for evidence combination • Illustrative examples of evidential reasoning
Email: jian-bo.yang@mbs.ac.uk, l.xu@mbs.ac.uk
Monday, December 16, 2013 Page 1
Main Topics of the Presentation
• An introduction to Bayesian inference • Generalised probabilistic reasoning and Dempster’s rule
Expert 2: "I am 40% sure it's meningitis and 30% sure it is concussion, but 90% sure it is NOT brain tumor and 40% sure it is NOT meningitis.” Expert 3: "I am 10% sure it's meningitis, 30% sure it is concussion, and 50% sure it is brain tumor.”
2
Bayesian Inference
Bayesian inference is a method of probabilistic reasoning in
which Bayes' rule is used to update the probability estimate
for a hypothesis as new evidence becomes available. It has the following features.
0
0.5 0
m1 m2
After normalisation
0
1
0
8
Main Topics of the Presentation
• An introduction to Bayesian inference • Generalised probabilistic reasoning and Dempster’s rule • Limitations of current probabilistic reasoning • Evidential reasoning rule for evidence combination • Illustrative examples of evidential reasoning