Reasoning Under Uncertainty

1
Reasoning Under Uncertainty

• Kostas Kontogiannis
• ECE 457

2
Uncertainty and Evidential Support

• In its simplest case, a Knowledge Base contains
  rules of the form
• A B C gt D
• where facts A, B, C are considered to be True
  (that is these facts hold with probability 1),
  and D is asserted in the Knowledge Base as being
  True (also with probability 1)
• However for realistic cases, domain knowledge
  has to be modeled in way that accommodates
  uncertainty. In other words we would like to
  encode domain knowledge using rules of the form
  
• A B C gt D
  (CFx1)
• where A, B, C are not necessarily certain
  (i.e. CF 1)

3
Issues in Rule-Based Reasoning Under Uncertainty

• Many rules support the same conclusion with
  various degrees of Certainty
• A1 A2 A3 gt H
  (CF0.5)
• B1 B2 B3 gt H
  (CF0.6)
• (If we assume all A1, A2, A3, B1, B3, B3 hold
  then H is supported with CF(H) CFcombine(0.5,
  0.6))
• The premises of a rule to be applied do not hold
  with absolute certainty (CF, or probability
  associated with a premise not equal to 1)
• Rule A1 gt H
  (CF0.5)
• However if during a consultation, A1 holds
  with CF(A1) 0.3 the H holds with CF(H)
  0.50.3 0.15

4
The Certainty Factor Model

• The potential for a single piece of negative
  evidence should not overwhelm several pieces of
  positive evidence and vice versa
• the computational expense of storing MBs and
  MDs should be avoided and instead maintain a
  cumulative CF value
• Simple model
• CF MB - MD
• Cfcombine(X, Y) X Y(1-X)
  
• The problem is that a single negative evidence
  overwhelms several pieces of positive evidence

5
The Revised CF Model

• MB - MD
• 1 - min(MB, MD)

CF

X Y(1 - X) X, Y gt 0
X Y
One of X, Y lt 0
CFcombine(X,Y)
1 - min(X, Y)
- CFcombine(-X, -Y) X, Y lt 0
6
Additional Use of CFs

• Provide methods for search termination
• A B C D E
• In the case of branching in the inference
  sequencing paths should be kept distinct

R1
R2
R3
R4
0.8
0.4
0.7
0.7
7
Cutoff in Complex Inferences
R4
D
E
R3
R1
0.7
R2
0.7
A
B
C
R5
0.8
0.4
F
0.9
We should maintain to paths for cutoff (0.2), one
being (E, D, C, B, A) and the other (F, C, B,
A). If we had one path then E, D, C would drop to
0.19 and make C unusable later in path F, C, B,
A.