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

T: States T and

E: States E and

T

C

E

Now, it may very well e.g. be that

Then C is said to have positive influence on T

Inequality sign of opposite direction ( ) negative influence

Pure equality sign (=) : no influence

pp HCTHCT ,Pr,Pr

H

T

C

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

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)

jiiixXYixXY xxyyFyFji

when allfor

What is the relationship with Bayesian networks?

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.

Qualitative Probability Networks (QPN)

1

3

H

T

C

E

2

4

Each takes one of the values “+” , “–”, “0” or “?”.

“+”: positive influence

“–”: negative influence

“0”: no influence

“?”: unknown influence

How can we assess e.g. the influence of E on H?

Sign product and addition operators:

Sign product, :

?"","","" and ?""

or ?"","","" and ?""if?""

"0"or "0"if"0"

"" and ""or "" and ""if""

"" and ""or "" and ""if""

ij

ji

ji

jiji

jiji

ji

otherwiseif?""

"0" and "0"if"0""" and "0", ""

or "0", "" and ""if""

"" and "0", ""

or "0", "" and ""if""

ji

ji

ji

ji

ji

ji

Sign addition :

1

3

H

T

C

E

2

4

Now, assume

1 = 2 = 3 = 4 = “+”

Let 5 be the unknown influence that E has on H

""

""

4

""

32

""

41

43215

1

3

H

T

C

E

2

4

What if

? ,Pr,Pr

and ,Pr,Pr

dp

dp

HCTHCT

HCTHCT

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. If disease A is present it is quite common to have an increased level of substance 1.

2. If disease B is present it is less common to have an increased level of substance 1.

3. 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

4. Recurrent fever attacks are quite common when disease B is present regardless of the level of substance 1

5. 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

6. 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

X

Y

Z

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”

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”

1

2

3

4

5

6

What influence has Z on H=A?

Synergy properties

Additive synergy:

A B

C

synergy zero

synergy negative

synergy positive

,Pr,Pr,Pr,Pr BACBACBACBAC

Product synergy:

synergy zero

synergy negative

synergy positive

,Pr

,Pr

,Pr

,Pr

BAC

BAC

BAC

BAC

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.