Chapter 7 Reasoning 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/

Outline

Introduction

Certainty Factor

Bayesian Reasoning

Dempster-Shafer Theory of Evidence

Summary

Uncertain agent

environmentagent

sensors

actuators

??

??

?

model

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

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

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)= -0.5cf(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)

otherwise

ECFifECFEHCFHCF

0

0)()(*),()( )0),(max{*),()( ECFEHCFHCF

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 RulesIF <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 RulesIF <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

Tuberculosis

Tuberculosisor Cancer

XRay Result Dyspnea

BronchitisLung Cancer

Smoking

Patient Information

Medical Difficulties

Diagnostic Tests

Medical DifficultiesTub 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

Dyspnea

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。

E

)|( EAP

网络

证据

查询

推理

)|( EAP

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

p(B)

B)p(A,B)|p(A

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

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

Components of DST

Basic probability assignment (m)

Belief and Plausibility measures

Let X be the universal set: the set of all states under consideration. 2X is the power set of X

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