Chapter 12 Certainty Theory (Evidential Reasoning) 1

Probability Theory Advantages: The oldest and best-established technique to deal with inexact knowledge and random data. works well in such areas where statistical data is usually available (with accurate probabilities) – e.g. forecasting and planning Disadvantages: reliable statistical information is not always available The conditional independence of evidence is not available in all real-world situations the Bayesian belief propagation is of exponential complexity,  impractical for large knowledge bases 2

3 Probability Theory Disadvantages (continued.) The conditional probabilities may be inconsistent with the prior probabilities given by the expert. – Because the expert made different assumptions when assessing the conditional and prior probabilities – Proved in pages , Durkin – Example and more explanation in pages 72-74, Negnevitsky Most human feel uncomfortable estimating probability, but are more willing to express their certainty (psychologically confirmed) – The probability of my catching a cold is 90%.  – I really believe that I’m catching a cold.

4 Bias of the Bayesian Method Consider, for example, a car that does not start and makes odd noises when you press the starter. The conditional probability of the starter being faulty if the car makes odd noises may be expressed as: IFthe symptomis "odd noises" THENthe starteris bad {with probability 0.7} p(starter is not bad | odd noises) = = p(starter is good | odd noises) = = 0.3

5 Bias of the Bayesian Method Therefore, we can obtain a companion rule that states IFthe symptomis “odd noises” THENthe starteris good {with probability 0.3} Domain experts do not deal with conditional probabilities and often deny the very existence of the hidden implicit probability (0.3 in our example). We would also use available statistical information and empirical studies to derive the following rules: IFthe starteris bad THENthe symptomis “odd noises” {probability 0.85} IFthe starteris bad THENthe symptomis not “odd noises” {probability 0.15}

6 Bias of the Bayesian Method To use the Bayesian rule, we still need the prior probability, the probability that the starter is bad if the car does not start. Suppose, the expert supplies us the value of 5 per cent. Now we can apply the Bayesian rule to obtain: The number obtained is significantly lower than the expert’s estimate of 0.7 given at the beginning of this section!!! The reason for the inconsistency is that the expert made different assumptions when assessing the conditional and prior probabilities.

7 Certainty Factors A simple approach & popular alternative to Bayesian reasoning. used in cases where the probabilities are not known or are too difficult or expensive to obtain better explanations of the control flow through a rule-based expert system lacks the mathematical correctness of the probability theory (the lack of a formal foundation) – The certainty theory is not “mathematically pure” but does mimic the thinking process of a human expert.

8 Certainty Factors & Probability Theory both share a common problem: finding an expert able to quantify personal, subjective and qualitative information. combined approach: PROSPECTOR – a geological expert system – probability-based knowledge base; but asking certainty measure from user – Example program using PROSPECTOR approach, pages , Durkin, Ch. 11.

Uncertain Terms 9

10 Certainty Factors The syntax of the rules in the knowledge base: IF THEN {cf } where cf represents belief in hypothesis H given that evidence E has occurred.

11 Certainty Factors MB: measure of belief MD: measure of disbelief The values of MB(H, E) and MD(H, E) range between 0 and 1. The total strength of belief or disbelief in a hypothesis:

12 Certainty Factors MB and MD can be calculated from the probabilities: p(H) is the prior probability of hypothesis H being true; p(H|E) is the probability that hypothesis H is true given evidence E. Q. Explain how MB & MD will change, when p(H|E) > p(H) and p(H|E) < p(H).

Evidential Reasoning The certainty factor assigned by a rule is propagated through the reasoning chain. Handling: – uncertain evidence – conjunctive rule, – disjunctive rule, – hypothesis with different degrees of belief 13

14 Evidential Reasoning: Uncertain Evidence IF E {cf(E)}, THEN H {cf} cf (H,E) = cf (E) x cf For example: IFsky is clear {cf 0.5} THENthe forecast is sunny {cf 0.8} cf (H,E) = 0.5 x 0.8 = 0.4 This result can be interpreted as "It may be sunny".

15 Evidential Reasoning: Conjunctive Rules Conjunctive Rules: cf (H, E 1  E 2 ...  E n ) = min [cf (E 1 ), cf (E 2 ),..., cf (E n )] x cf For example: IFskyis clear {cf 0.9} ANDthe forecastis sunny {cf 0.7} THENthe actionis 'wear sunglasses' {cf 0.8} cf (H, E 1  E 2 ) = min [0.9, 0.7] x 0.8 = 0.7 x 0.8 = 0.56

16 Evidential Reasoning: Disjunctive Rules Disjunctive Rules: cf (H, E1  E2 ...  E n ) = max [cf (E 1 ), cf (E 2 ),..., cf (E n )] x cf For example: IFskyis overcast {cf 0.6} ORthe forecastis rain {cf 0.8} THENthe actionis 'take an umbrella' {cf 0.9} cf (H, E 1  E 2 ) = max [0.6, 0.8] x 0.9 = 0.8 x 0.9 = 0.72

17 Evidential Reasoning: Hypothesis with different degrees of belief Example: Rule 1:IFA is X THENC is Z {cf 0.8} Rule 2:IFB is Y THENC is Z {cf 0.6} If both Rule 1 and Rule 2 are fired, what certainty should be assigned to object C having value Z? Combining certainty factors for a hypothesis.

18 Evidential Reasoning: Hypothesis with different degrees of belief where: cf 1 is the confidence in hypothesis H established by Rule 1; cf 2 is the confidence in hypothesis H established by Rule 2; |cf 1 | and |cf 2 | are absolute magnitudes of cf 1 and cf 2, respectively.

19 FORECAST an application of certainty factors

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FORECAST an application of certainty factors 21

FORECAST an application of certainty factors 22

FORECAST an application of certainty factors 23