Reasoning Under Uncertainty Kostas Kontogiannis ECE 457 Uncertainty

Uncertainty and Evidential Support • In its simplest case, a Knowledge Base contains rules

Issues in Rule-Based Reasoning Under Uncertainty • Many rules support the same conclusion with

The Certainty Factor Model • The potential for a single piece of negative evidence

The Revised CF Model CF = MB - MD 1 - min(MB, MD) {

Additional Use of CFs • Provide methods for search termination A R 1 0.

Cutoff in Complex Inferences A R 1 0. 8 B R 2 0. 4

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Reasoning Under Uncertainty Kostas Kontogiannis E&CE 457

Uncertainty and Evidential Support • In its simplest case, a Knowledge Base contains rules of the form : A & B & C => 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 => D (CF: x 1) where A, B, C are not necessarily certain (i. e. CF = 1)

Issues in Rule-Based Reasoning Under Uncertainty • Many rules support the same conclusion with various degrees of Certainty A 1 & A 2 & A 3 => H (CF=0. 5) B 1 & B 2 & B 3 => H (CF=0. 6) (If we assume all A 1, A 2, A 3, B 1, B 3 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: A 1 => H (CF=0. 5) However if during a consultation, A 1 holds with CF(A 1) = 0. 3 the H holds with CF(H) = 0. 5*0. 3 = 0. 15

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 MB’s and MD’s 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

The Revised CF Model CF = MB - MD 1 - min(MB, MD) { CFcombine(X, Y) = X + Y(1 - X) X, Y > 0 X+Y One of X, Y < 0 1 - min(|X|, |Y|) - CFcombine(-X, -Y) X, Y < 0

Additional Use of CFs • Provide methods for search termination A R 1 0. 8 B R 2 0. 4 C R 3 0. 7 D R 4 E 0. 7 • In the case of branching in the inference sequencing paths should be kept distinct

Cutoff in Complex Inferences A R 1 0. 8 B R 2 0. 4 R 3 C D 0. 9 E 0. 7 R 5 R 4 F 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.