Chapter 7 Reasoning in Uncertain Situations

1. Chapter 7Reasoning in Uncertain Situations Xiu-jun GONG (Ph. D) School of Computer Science and Technology, Tianjin University gongxj@tju.edu.cn http://cs.tju.edu.cn/faculties/gongxj/course/ai/

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

3. sensors ? agent actuators model Uncertain agent ? 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 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

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

17. p(A, B) p(B | A)p(A) = = p(A | B) p(B) p(B) 贝叶斯推理可以在反复使用贝叶斯规则而获得 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

18. Inference Methods • Exact 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 • Ignorance: toss a coin for probability • 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

20. Components of DST 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