Transcript Qualitative probability models

Qualitative probability models
How can we handle cases where explicit probabilities
cannot be assigned (neither based on experience, nor as
estimates from historical record) ?
Might still be feasible to order probabilities.
 Qualitative relationships
Standard transfer
network:
H: States Hp and Hd
C: States C and C
T: States T and T
E: States E and E
H
Now, it may very well e.g. be that
C
T



Pr T C , H p  Pr T C , H p

Then C is said to have positive influence on T
Inequality sign of opposite direction ( ) 
negative influence
Pure equality sign (=) : no influence
E
Binary case (like above): Simple.
Non-binary case: More involved. The use of the cumulative
distributiíon function sorts things out:
If for two variables X and Y
FY X  xi  yi   FY X  x  yi  for all yi when xi  x j
j
then X is said to have positive influence on Y (negative influence if the
direction of the inequality sign is the opposite, and no influence if it is
an equal sign)
Note! Symmetry property:
If for any types of variables (X and Y ) X has positive (negative)
influence on Y then Y has positive (negative) influence on X.
What is the relationship with Bayesian networks?
Qualitative Probability Networks (QPN)
H
2
1
Each  takes one of the values “+” , “–”,
“0” or “?”.
C
T
3
“+”: positive influence
“–”: negative influence
“0”: no influence
“?”: unknown influence
4
E
How can we assess e.g. the influence of E on
H?
Sign product and addition operators:
Sign product, :
""
""

 i   j  "0"

" ?"

if
if
if
if
i
i
i
i
"" and  j "" or  i "" and  j ""
"" and  j "" or  i "" and  j ""
"0" or  j "0"
" ?" and  j  "" , "" , " ?" or
 j " ?" and  i  "" , "" , " ?"
Sign addition :

""


 i   j  ""

"0"

" ?"
if
if
 i "" and  j  "" , "0" or
 i  "" , "0" and  j ""
 i "" and  j  "" , "0" or
if
 i  "" , "0" and  j ""
 i "0" and  j "0"
if
otherwise
H
Now, assume
2
1
1 = 2 = 3 = 4 = “+”
C
T
3
Let 5 be the unknown influence that E
has on H
 5  1   2   3    4 
 1   4   2   3   4 ""

4
" "

" "


" "
E
What if
H

2
1
C
T
4
E
3

Pr T C , H p  Pr T C , H d  and

 
Pr T C , H p  Pr T C , H d

?
Example: In the head of the experienced examiner revisited
Assume there is a question whether an individual has a specific disease A or
another disease B.
What is observed is
The individual has an increased level of substance 1
The individual has recurrent fever attacks
The individual has light recurrent pain in the stomach
The experience of the examining physician says
1.
2.
3.
4.
5.
6.
If disease A is present it is quite common to have an increased level
of substance 1.
If disease B is present it is less common to have an increased level
of substance 1.
If disease A is present it is not generally common to have recurrent
fever attacks, but if there is also an increased level of substance 1
such events are very common
Recurrent fever attacks are quite common when disease B is
present regardless of the level of substance 1
Recurrent pain in the stomach are generally more common when
disease B is present than when disease A is present, and regardless
of the level of substance 1 and whether fever attacks are present or
not
If a patient has disease A, increased levels of substance 1 and
recurrent fever attacks he/she would almost certainly have
recurrent pain in the stomach. Otherwise, if disease A is present
recurrent pain in the stomach is equally common.
H
H:
A: “Disease A”
B: “Disease B”
X:
x1 : “The individual has an increased level of
substance 1”
x2 : “The individual has a normal level of
substance 1”
1
2
3
X
5
Y
4
6
Z
Y:
y1 : “The individual has recurrent fever attacks”
y2 : “The individual has no fever attacks”
Z:
z1 : “The individual has light recurrent pain in the
stomach”
z2 : “The individual has no pain in the stomach”
What influence has Z on H=A?
Synergy properties
A
B
C
Additive synergy:

Pr C A, B   Pr C A, B


 Pr C A, B  Pr C A, B


 

positive synergy
negative synergy
zero synergy
Product synergy:
Pr C A, B 

Pr C A, B




 
Pr C A, B 
Pr C A, B
positive synergy
negative synergy
zero synergy
One specific use of product synergy:
If there is a negative product synergy between two binary parental
nodes then confirmation of the positive state of one of them reduces
the belief of a positive state of the other. Explaining away
If there is a positive product synergy, then confirmation of the positive
state of one of them increases the belief of a positive state of the other.