1. Chapter 7 Reasoning in Uncertain Situations
Xiu-jun GONG (Ph. D)
School of Computer Science and Technology, Tianjin University

2. Outline Introduction Certainty Factor Bayesian Reasoning
Dempster-Shafer Theory of Evidence
Summary

3. Uncertain agent
sensors
actuators ?
agent ?
model
environment ?

4. Types of Uncertainty Uncertainty in prior knowledge
E.g., some causes of a disease are unknown and are not represented in the background knowledge of a medical-assistant agent
Uncertainty in actions
Actions are represented with relatively short lists of preconditions, while these lists are in fact arbitrary long. It is not efficient (or even possible) to list all the possibilities.
Uncertainty in perception
E.g., sensors do not return exact or complete information about the world; a robot never knows exactly its position.

5. Sources of uncertainty
Epistemic uncertainty：subjective uncertainty
Aleatory uncertainty： Objective uncertainty
Aleatory[ˈeiliətəri]: 偶然
Epistemic[ˌepiˈsti:mik]： 认识
What we call uncertainty is a summary of all that is not explicitly taken into account in the agent’s knowledge base.

6. Questions How to represent uncertainty in knowledge?
How to perform inferences with uncertain knowledge?
Which action to choose under uncertainty?

7. Uncertainty Approaches in AI
Quantitative
Probability Theory & Fuzzy logic
Certainty Factors
Bayesian Inference
Dempster-Shafer evidence theory
Qualitative
Logical Approaches
Reasoning by cases
Non-monotonic reasoning
Hybrid approaches

8. Certainty Factors Certainty factors express belief in an event
Fact or hypothesis
Based upon evidence
Experts assessment
Composite number that can be used to
Guide reasoning
Cause a current goal to be deemed unpromising and pruned from search space
Rank hypotheses after all evidence has been considered

9. Certainty Factors for Evidence
Certainty Factor cf(E) is a measure of how confident we are in E
Range from –1 to +1
cf=-1 very uncertain
cf=+1 very certain
cf=0 neutral
Certainty factors are relative measures
Do not translate to measure of absolute belief
First introduced in an expert system called MYCIN for the diagnosis and therapy of blood infections

10. CF for rules
Certainty factors combine belief and disbelief into a single number based on some evidence
MB(H,E)-measure of belief in H given evidence E
MD(H,E)-measure of disbelief in H given evidence E
Strength of belief or disbelief in H depends on the kind of evidence E observed
cf(H,E)= MB(H,E) – MD(H,E)

11. Belief
Positive CF implies evidence supports hypothesis since MB > MD
CF of 1 means evidence definitely supports the hypothesis
CF of 0 means either there is no evidence or that the belief is cancelled out by the disbelief
Negative CF implies that the evidence favours negation of hypothesis since MB < MD

12. Stanford CF Algebra
There are rules to combine CFs of several evidences
CF (E1 and E2) = MIN { CF(E1) , CF(E2) }
CF (E1 or E2) = MAX { CF(E1) , CF(E2) }
cf(shep is a dog)=0.7
cf(shep has wings)=-0.5
cf(Shep is a dog and has wings) = min(0.7, -0.5)
= -0.5
cf(Shep is a dog or has wings) = max(0.7, -0.5)
= 0.7

13. CF Inference Known CF(E) and CF(H,E), solve for CF(H)
Ex1: CF(cold，fever)=0.6, CF(fever)=0.7
then CF (cold)=0.6 * 0.7 =0.42
Ex2: CF(cold，fever)=0.6，CF(fever)= - 0.8
then CF (cold)=0

14. CF Conjunctive Rules cf(H, E1  E2  …  En) =
IF <evidence1>
AND <evidence2> .
AND <evidencen>
THEN
<hypothesis H> {cf}
cf(H, E1  E2  …  En) =
min[cf(E1),cf(E2)…cf(En)] x cf

15. CF: Disjunctive Rules cf(H, E1  E2  …  En) =
IF <evidence1>
OR <evidence2> .
OR <evidencen>
THEN
<hypothesis H> {cf}
cf(H, E1  E2  …  En) =
max[cf(E1),cf(E2)…cf(En)] x cf

16. Bayesian Network Visit Asia Smoking Tuberculosis Lung Cancer
Patient Information
Tuberculosis
Lung Cancer
Medical Difficulties
Tub or Can
True
False
Bronchitis
Present
Absent
0.90
0.70
0.80
0.10
0.l0
0.30
0.20
Dyspnea
Bronchitis
Medical Difficulties
Tuberculosis
or Cancer
XRay Result
Dyspnea
Diagnostic Tests

17. Evidence Reasoning in BN
目的：通过联合概率分布公式，在给定的网络结构
和已知证据下，计算某一事件的发生的概率。 网络 推理 证据 E 查询
贝叶斯推理可以在反复使用贝叶斯规则而获得 =
p(B)
A)p(A) |
p(B B)
p(A,
p(A

18. Inference Methods Exact reasoning Approximate reasoning
网络的拓扑结构是推理复杂性的主要原因；
当前的一些精确算法是有效地，能够解决现实中的大部分问题
由于对知识的认知程度，精确推理还存在一些问题
Approximate reasoning
证据的低似然性和函数关系 是近似推理中复杂性的主要原因

19. Dempster-Shafer Theory
The D-S theory is a mathematical theory of evidence based on belief functions and plausible reasoning
Why we need D–S theory
Ignorance: toss a coin for probability
No information for the coin: 0.5 head, 0.5 tail
Known that the coin is fair: 0.5 head, 0.5 tail
No information for the coin: B(H)=0, B(T)=0.
Known that the coin is fair: B(H)=0.5, B(T)=0.5.
Probabilities on a set which is related to a set of situations
plausible reasoning: 拟真推理,似然推理
Example:The first gamble is that we bet on a head turning up when we toss a coin that is known to be fair. Now consider the second gamble, in which we bet on the outcome of a fight between the world's greatest boxer and the world's greatest wrestler. Assume we are fairly ignorant about martial arts and would have great difficulty making a choice of who to bet on.

20. Components of DST Basic probability assignment (m)
Let X be the universal set: the set of all states under consideration. 2X is the power set of X
Basic probability assignment (m)
Belief and Plausibility measures

21. Rules of Combination
 to combine two independent sets of mass assignments m1,m2:
Where K is a measure of the amount of conflict between the two mass sets

22. Discussion on D-S
Many of the criticisms of D-S have rejected the theory on the basis of the Dempster rule of combination.
There are many ways to combine evidence in D-S.
The critical concern for the selection of a combination operation is the nature of conflict and how it is handled by a particular combination rule.

23. Summary Most of situations are uncertainty in nature Certainty Factor
Bayesian Reasoning
Dempster-Shafer Theory of Evidence