›
›

Bài giảng Trí tuệ nhân tạo - Chapter 9: Uncertain KR&R

Download
Uncertain KR&R  
Chapter 9  
Outline  
• Probability  
• Bayesian networks  
• Fuzzy logic  
2
Probability  
FOL fails for a domain due to:  
Laziness: too much to list the complete set of rules, too hard to  
use the enormous rules that result  
Theoretical ignorance: there is no complete theory for the  
domain  
Practical ignorance: have not or cannot run all necessary tests  
3
Probability  
• Probability = a degree of belief  
• Probability comes from:  
Frequentist: experiments and statistical assessment  
Objectivist: real aspects of the universe  
Subjectivist: a way of characterizing an agent’s beliefs  
• Decision theory = probability theory + utility theory  
4
Probability  
Prior probability: probability in the absence of any other  
information  
P(Dice = 2) = 1/6  
random variable: Dice  
domain = <1, 2, 3, 4, 5, 6>  
probability distribution: P(Dice) = <1/6, 1/6, 1/6, 1/6, 1/6, 1/6>  
5
Probability  
Conditional probability: probability in the presence of  
some evidence  
P(Dice = 2 | Dice is even) = 1/3  
P(Dice = 2 | Dice is odd) = 0  
P(A | B) = P(A ∧ B)/P(B)  
P(A ∧ B) = P(A | B).P(B)  
6
Probability  
Example:  
S = stiff neck  
M = meningitis  
P(S | M) = 0.5  
P(M) = 1/50000  
P(S) = 1/20  
P(M | S) = P(S | M).P(M)/P(S) = 1/5000  
7
Probability  
Joint probability distributions:  
X: <x1, …, xm> Y: <y1, …, yn>  
P(X = xi, Y = yj)  
8
Probability  
Axioms:  
• 0 ≤ P(A) ≤ 1  
• P(true) = 1 and P(false) = 0  
• P(A ∨ B) = P(A) + P(B) − P(A ∧ B)  
9
Probability  
Derived properties:  
• P(¬A) = 1 − P(A)  
• P(U) = P(A1) + P(A2) + ... + P(An)  
U = A1 ∨ A2 ∨ ... ∨ An collectively exhaustive  
Ai ∧ Aj = false  
mutually exclusive  
10  
Probability  
Bayes’ theorem:  
P(Hi | E) = P(E | Hi).P(Hi)/∑iP(E | Hi).P(Hi)  
Hi’s are collectively exhaustive & mutually exclusive  
11  
Probability  
Problem: a full joint probability distribution P(X1, X2, ..., Xn)  
is sufficient for computing any (conditional) probability on  
Xi's, but the number of joint probabilities is exponential.  
12  
Probability  
• Independence:  
P(A ∧ B) = P(A).P(B)  
P(A) = P(A | B)  
• Conditional independence:  
P(A ∧ B | E) = P(A | E).P(B | E)  
P(A | E) = P(A | E ∧ B)  
Example:  
P(Toothache | Cavity ∧ Catch) = P(Toothache | Cavity)  
P(Catch | Cavity ∧ Toothache) = P(Catch | Cavity)  
13  
Probability  
"In John's and Mary's house, an alarm is installed to  
sound in case of burglary or earthquake. When the alarm  
sounds, John and Mary may make a call for help or  
rescue."  
14  
Probability  
"In John's and Mary's house, an alarm is installed to  
sound in case of burglary or earthquake. When the alarm  
sounds, John and Mary may make a call for help or  
rescue."  
Q1: If earthquake happens, how likely will John make a  
call?  
15  
Probability  
"In John's and Mary's house, an alarm is installed to  
sound in case of burglary or earthquake. When the alarm  
sounds, John and Mary may make a call for help or  
rescue."  
Q1: If earthquake happens, how likely will John make a  
call?  
Q2: If the alarm sounds, how likely is the house  
burglarized?  
16  
Bayesian Networks  
• Pearl, J. (1982). Reverend Bayes on Inference  
Engines: A Distributed Hierarchical Approach,  
presented at the Second National Conference on Artificial  
Intelligence (AAAI-82), Pittsburgh, Pennsylvania  
• 2000 AAAI Classic Paper Award  
17  
Bayesian Networks  
P(B)  
P(E)  
.002  
Burglary  
Earthquake  
.001  
B
E
P(A)  
T
T
F
F
T .95  
F .94  
T .29  
F .001  
Alarm  
A
T
F
P(M)  
.70  
A
T
F
P(J)  
.90  
MaryCalls  
JohnCalls  
.01  
.05  
18  
Bayesian Networks  
Syntax:  
• A set of random variables makes up the nodes  
• A set of directed links connects pairs of nodes  
• Each node has a conditional probability table that quantifies the  
effects of its parent nodes  
• The graph has no directed cycles  
A
B
D
A
B
D
yes  
no  
C
C
19  
                                        
                                        
                                                
                                                
                                                                 
                                                                 
                                            
                                            
                                                      
                                                      
                                                        
                                                        
                                                         
                                                          
                                                                          
                                                                          
                                                                                         
                                                                                         
                                                                                           
                                                                                           
                                                                                            
                                                                                            
                                                                                                     
                                                                                                     
                                                                                                       
                                                                                                       
                                                                                                        
                                                                                                        
                                                                                                                         
                                                                                                                         
                                                                                                                                                 
                                                                                                                                                 
                                                                                                                                                           
                                                                                                                                                           
                                                                                                                                                                         
                                                                                                                                                                         
                                       
                                        
                                                 
                                                 
                                                             
                                                             
                                                              
                                                              
                                                               
                                                               
                                                                               
                                                                               
                                                                                         
                                                                                         
                                                                                                  
                                                                                                  
                                                                                                                               
                                                                                                                                
                                          
                                          
                                                   
                                                   
                                                   
                                                    
                                                     
                                                     
                                                                     
                                                                     
                                                                                       
                                                                                       
                                                                                                                  
                                                                                                                  
                                                                                                                                                      
                                                                                                                                                      
                                                                                                                                                                    
                                                                                                                                                                    
                                                                                                                                                                     
                                                                                                                                                                     
                                                                                                                                                                      
                                                                                                                                                                      
                                                                                                                                                                                      
                                                                                                                                                                                      
Bayesian Networks  
Semantics:  
• An ordering on the nodes: Xi is a predecessor of Xj ⇒ i < j  
• P(X1, X2, …, Xn)  
= P(Xn | Xn-1, …, X1).P(Xn-1 | Xn-2, …, X1). … .P(X2 | X1).P(X1)  
= ΠiP(Xi | Xi-1, …, X1) = ΠiP(Xi | Parents(Xi))  
P (Xi | Xi-1, …, X1) = P(Xi | Parents(Xi)) Parents(Xi) ⊆ {Xi-1, …, X1}  
Each node is conditionally independent of its predecessors given its  
parents  
20  
Tải về để xem bản đầy đủ
pdf 72 trang myanh 19080 Free
Download
Bạn đang xem 20 trang mẫu của tài liệu "Bài giảng Trí tuệ nhân tạo - Chapter 9: Uncertain KR&R", để tải tài liệu gốc về máy hãy click vào nút Download ở trên

File đính kèm:

  • pdfbai_giang_tri_tue_nhan_tao_chapter_9_uncertain_krr.pdf