# Chapter 7 Reasoning in Uncertain Situations

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

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

Uncertain agent sensors actuators ? agent ? model environment ?

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.

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.

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

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

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

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

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)

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

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

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

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

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

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

Evidence Reasoning in BN

Inference Methods Exact reasoning Approximate reasoning

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.

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

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

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.

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

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