Chapter 7
Reasoning in Uncertain Situations
Xiu-jun GONG (Ph. D)
School of Computer Science and Technology, Tianjin
University
[email protected]
http://cs.tju.edu.cn/faculties/gongxj/course/ai/
Outline

Introduction

Certainty Factor

Bayesian Reasoning

Dempster-Shafer Theory of Evidence

Summary
Uncertain agent
sensors
?
?
environment
agent
?
actuators
model
Types of Uncertainty

Uncertainty in prior knowledge


Uncertainty in actions


E.g., some causes of a disease are unknown and are not
represented in the background knowledge of a medicalassistant agent
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


Based upon evidence


Fact or hypothesis
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)
= -0.5
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)
CF ( H , E ) * CF ( E ) if CF ( E )  0
CF ( H )  
0 otherwise
CF ( H )  CF ( H , E ) * max{ CF ( E ),0)
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
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
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
Patient Information
Tuberculosis
Lung Cancer
Tuberculosis
or Cancer
XRay Result
Bronchitis
Dyspnea
Medical
Difficulties
Medical
Difficulties
Present
Absent
Tub or Can
Bronchitis
True
Present
0.90
0.l0
True
Absent
0.70
0.30
False
Present
0.80
0.20
False
Absent
0.10
0.90
Dyspnea
Diagnostic Tests
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
证据
查询
P( A | E )
推理
E
P( A | E )
p(A | B) 
p(A, B)
p(B)

p(B | A)p(A)
p(B)
贝叶斯推理可以在反复使用贝叶斯规则而获得
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



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
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
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