Stanford Certainty factor Algebra

For a hypothesis (or rule) H:

• Definitions:

MB(H|E) = measure of belief in H given evidence E
MD(H|E) = measure of disbelief in H given evidence E

• Assumptions:

(1 >= MB(H|E) >= 0) and (MD(H|E) == 0) OR
(1 >= MD(H|E) >= 0) and (MB(H|E) == 0)

• Definition:

CF(H|E) = MB(H|E) - MD(H|E)

Combining Certainty Factors:

• Dynamic Certainty for a Conclusion:

Given a rule R = "if H then C with certainty CF(R)" and a calculated certainty CF(H),
CF(C) = CF(H) x CF(R)

• Conjunction and Disjunction of Hypotheses:

CF (P1 and P2) = MIN { CF(P1) , CF(P2) }
CF (P1 or P2) = MAX { CF(P1) , CF(P2) }

• Multiple Rules with Same Conclusion:

Suppose we have two rules R1 and R2 with the same conclusion C,
CF(C) = CF(R1) + CF(R2) - (CF(R1) x CF(R2)), when both CF(R1) and CF(R2) are positive
CF(C) = CF(R1) + CF(R2) + (CF(R1) x CF(R2)), when both CF(R1) and CF(R2) are negative
(CF(R1) + CF(R2)) / (1 - MIN { |CF(R1)| , |CF(R2)| }), when the signs are different

Example:

Rule Base
--------
 R1: if a and b then x  (cf = 0.5)
 R2: if c or d then x   (cf = 0.7)

Dynamic Input:
-------------
 a, with certainty 1.0
 b, with certainty 0.8
 c, with certainty 0.9
 d, with certainty 0.7

Compute CF values for x:
-----------------------
CF(a and b) = MIN{1.0, 0.8} = 0.8
 ==> CF1(x) = 0.8 * 0.5 = 0.4

CF(c or d) = MAX{0.9, 0.7} = 0.9
 ==> CF2(x) = 0.9 * 0.7 = 0.63

CF(x) = 0.4 + 0.63 - 0.4*0.63 = 0.778  

Closed Form