Bai giai tri tue nhan tao tut 7

TRƯỜNG ĐẠI HỌC BÁCH KHOA TP.HCM

Khoa Khoa học & Kỹ thuật Máy tính

1

ARTIFICIAL INTELLIGENCE

Tutorial 7 Questions

UNCERTAINTY and IMPRECISION

Question 1.

1. Prove that:

a. P(A | B ˄ A) = 1

b. P(X ˄ Y | E) = P(X | Y ˄ E).P(Y | E)

c. P(Y | X ˄ E) = P(X | Y ˄ E).P(Y | E)/P(X | E)

(the conditionalized version of Bayes' rule)

Solution:

a. 1

( )

( )

( )

( ( )) ( | )  

  

 

  P B A

P B A

P B A

P A B A P A B A

b.

( )

( ) ( | ) P E

P X Y E P X Y E  

 

c.

( ) ( | ). ( | ). ( ) ( | ). ( | ) (| ) ( ) ( | ). ( ) ( | )

PY X E P X Y E PY E PE P X Y E PY E PY X E

PX E PX E PE PX E

  

  



2. Show that the statement

P(A ˄ B | C) = P(A | C).P(B | C)

is equivalent to either of the statements

P(A | B ˄ C) = P(A | C) and P(B | A ˄ C) = P(B | C)

Solution:

( | ). ( | ) ( )

( | ). ( )

( )

( ) ( | ) P A B C P B C

P C

P A B C P B C

P C

P A B C P A B C  

    

 

Therefore, if P(A B |C)  P(A|C).P(B |C) then P(A| B C)  P(A|C)

Similarly, if P(A B |C)  P(A|C).P(B |C) then P(B | A  C)  P(B | C)