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Dealing with uncertainty requires reasoning under uncertainty along with possessing a lot of common sense. Different strategies or theories have been proposed for handling uncertainty in expert systems. The basis for selection depends on the nature of uncertainty. Some of the theories include; Bayesian Probability, Certainty Factor Theory, Hartley Theory, Shannon Theory, Dempster-Shafer Theory, Markov Models and Zadeh‘s Fuzzy Theory(Dubey et al., 2014). Bayesian Probability Reasoning and Certainty Factors Theory shall be discussed briefly.

(a) Bayesian Reasoning

Bayes‘s theorem is a mechanism for combining new andexistent evidence, usually given as subjective probabilities. Itis used to revise existing prior probabilities based on newinformation. The Bayesian approach is based on subjectiveprobabilities (i.e., probabilities estimated by a

managerwithout the benefit of a formal model); a subjectiveprobability is provided for each proposition.

Suppose all rules in the knowledge base are represented in the following form in equation (2.36):

IF H is true

THEN E is true {with probability p} (2.36)

This rule implies that if event H occurs, the the probability that event E will occur is p. In expert systems, H usually represents a hypothesis and E denotes evidence to support this hypothesis. The Bayesian rule expressed in terms of Hypotheses and evidence looks like equation (2.37):

𝑝(H|E) =

𝑝(E|H) ∗ 𝑝(H) + 𝑝(E|¬H) ∗ 𝑝(¬H) (2.37)

wherep(H) is the prior probability of hypothesis H being true;

p(E|H) is the probability that hypothesis H being true will result in evidence E;

p(¬H) is the prior probability of hypothesis H being false;

p(E|¬H) is the probability of finding evidence E even when hypothesis H is false.

In expert systems, the probabilities required to solve a problem are provided by experts. An expert determines the prior probabilities for possible hypotheses p(H) and p(¬H) , and also the conditional probabilities for observing evidence E if hypothesis H is true, p(E|H), and if hypothesis H is false, p(E|¬H).

Users provide provide information about the evidence observed and the expert system computes p(H|E) for hypothesis H in the light of the user-supplied evidence E. Probability p(H|E) is called the posterior probability of hypothesis h upon observing evidence E (Negnevitsky, 2002), (Dubey et al.,2014).

The bias of the Bayesian method is that the framework for Bayesian reasoning requires probability values as primary inputs. The assessment of these values usually involves human judgment.

However, psychological research shows that humans either cannot elicit probability values consistent with the Bayesian rules. This suggests that the conditional probabilities may be

inconsistent with the prior probabilities given by the expert. Domain experts do not deal with conditional probabilities and often deny the very existence of the hidden implicit probability (Negnevitsky, 2002).

(b) Certainty Factors Theory and Evidential Reasoning

Certainty theory is a framework for representing andworking with degrees of belief of true and false inknowledge-based systems. In certainty theory as in fuzzylogic, uncertainty is represented as a degree of belief. Thereare two steps in using every nonprobabilistic method ofuncertainty. First, it is necessary to be able to express thedegree of belief. Second, it is necessary to manipulate (e.g.,combine) degrees of belief when using knowledge-basedsystems. Certainty theory relies on the use of certaintyfactors. Certainty factors (CF) express belief in an event (or afact or a hypothesis) based on evidence (or on the expert‘sassessment). There are several methods of using certaintyfactors to handle uncertainty in knowledge-based systems.One way is to use 1.0 or 100 for absolute truth (i.e., completeconfidence) and 0 for certain falsehood. Certainty factors arenot probabilities. For example, when we say there is a 90percent chance of rain, there is either rain (90 percent) or norain (10 percent)(Dubey et al., 2014). In a non probabilistic approach, we can saythat a certainty factor of 90 for rain means that it is very likelyto rain. It does not necessarily mean that we express anyopinion about our argument of no rain (which is notnecessarily 10).Thus, certainty factors do not have to sum upto 100. Certainty theory introduces the concepts of belief anddisbelief (i.e., the degree of belief that something is not goingto happen).These concepts are independent of each other andso cannot be combined in the same way as probabilities(Negnevitsky, 2002), (Dubey et al.,2014).

Another assumption of certainty theory is that the knowledge content of rules is much more important than the algebra ofconfidences that holds the system together. Confidence measures correspond to the information evaluations that human experts attach to their conclusions (e.g., ―It is probably true‖ or ―It is highly unlikely‖). Certainty factors represent information about how certain the conclusion in a rule may be. Certainty factors can be attached both to the conditions in an if-then rule and to its conclusion. They aread hoc values, given by the experts based on experience or bythe

users when providing initial data. Certainty factors arenot probabilities, they represent beliefs about how strong given evidence is and to what degree the evidence supports ahypothesis. Certainty factors are measured using variousscales bot numeric (0 – 100, 0 – 10, 0 – 1, -1 to 1) andlinguistics ones (certain, fairly certain, likely, unlikely,highly unlikely, definitely not), see Table 2.2. It is used to judge uncertain evidence or conclusion. It deals with evidence in terms of their belief ordisbelief of each hypothesis (Dubey et al., 2014).

Table 2.2 Uncertain Terms and their Interpretation (Negnevitsky, 2005)

Term Certainty Factor

Definitely not -1.0

Almost certainly not -0.8

Probably not -0.6

Maybe not -0.4

Unknown -0.2 to +0.2

Maybe +0.4

Probably +0.6

Almost certainly +0.8

Definitely +1.0

 Higher certainty factors indicate strong confidence in a hypothesis.

 Certainty factors that approach -1 indicate confidence against a hypothesis.

 Certainty factors around 0 mean that we don‘t have information either for or against a hypothesis.

As the certainty factor (CF) approaches 1, the evidence isstronger for a hypothesis; as CF approaches - 1, theconfidence against the hypothesis gets stronger; and a CFaround 0 indicates that either little evidence in the rule'sreliability. Certainty measures may be adjusted to tune thesystem's performance, although slight variations in theconfidence measure tend to have little effect on the overall ofthe system. This second role of certainty measures confirmsthe belief that "the knowledge gives the power", than is, theintegrity of the knowledge itself best supports the productionof correct diagnoses (Isram andSimri, 2013).

In expert systems with certaintyfactors, the knowledge base consists of a set of rules that havethe following syntax in equation (2.38):

IF <evidence>

THEN <hypothesis> {cf} (2.38)

where cf represents belief in hypothesis, H given that evidence, E has occurred.

The certainty factors theory is based on twofunctions: measure of belief MB(H,E) and measure ofdisbelief MD(H,E).The values of MB(H,E) and MD(H,E) range between 0 and 1 in equations (2.39 and 2.40).

{ 1 if p(H) = 1

MB(H,E) = {max [𝑝(H|E),𝑝(H)] − 𝑝(H)

max [1,0] − 𝑝(H) otherwise (2.39)

{ 1 if p(H) = 0

MD(H,E) = { min [𝑝(H|E),𝑝(H)] − 𝑝(H)

min [1,0] − 𝑝(H) otherwise (2.40) Where 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;

The strength of belief or disbelief in hypothesis, H depends on the kind of evidence, E observed.

Some facts may increase the strength of belief, but some increase the strength of disbelief. The total strength of belief or disbelief in a hypothesis is as expressed in equation (2.41) (Negnevitsky, 2002):

𝑐𝑓 =

1 – min [MB (H,E),MD (H,E)] (2.41)

The certainty factor assigned by a rule is propagated through the reasoning chain. This involves establishing the net certainty of the rule consequent when the evidence in the rule antecedent is uncertain given by equation (2.42):

cf(H,E) = cf(E) * cf (2.42) For example,

IF sky is clear

THEN the forecast is sunny {cf 0.8}

and the current certainty factor of ―sky is clear” is 0.5, then cf(H,E) = 0.5 * 0.8 = 0.4 . This result can be interpreted as ―It may be sunny‖.

The certainty factor for conjunctive rules such as equation (2.43):

IF <evudence, E1>

. . .

AND < evidence, E_n>

THEN <hypothesis, H> {cf} (2.43)

the certainty of hypothesis H, is established as follows in equation (2.44):

cf(H,E₁∩E2∩…∩ En) = min [cf(E₁), cf(E₂), …, cf(En)] * cf (2.44) For example,

IF sky is clear

AND the forecast is sunny

THEN the action is ‗wear sunglasses’ {cf 0.8}

and the current certainty factor of ―sky is clear” is 0.9 and the certainty of the forecast of sunny is 0.7, then

cf(H,E1∩E2) = min[0.9, 0.7] * 0.8 = 0.7 * 0.8 = 0.56

The certainty factor for disjunctive rules such as equation (2.45):

IF <evudence, E1>

. . .

OR < evidence, En>

THEN <hypothesis, H> {cf} (2.45)

the certainty of hypothesis H, is established as follows in equation (2.46):

cf(H,E1UE2U … UEn) = max [cf(E1), cf(E2), …, cf(En)] * cf (2.46) For example,

IF sky is overcast OR the forecast is rain

THEN the action is ‗take an umbrella‘ {cf 0.9}

and the current certainty factor of ―sky is clear” is 0.6 and the certainty of the forecast of rain is 0.8, then

cf(H,E₁UE₂) = max[0.6, 0.8]* 0.9 = 0.8 * 0.9 = 0.72

Certainty factors are used if theprobabilities are not known or cannot be easily obtained.Certainty theory can manage incrementally acquired evidence, the conjunction and disjunction of hypotheses, aswell as evidences with different degrees of belief. Althoughthe certainty factors approach lacks the mathematicalcorrectness of the probability theory, it outperformssubjective Bayesian reasoning in such areas as diagnostics.Certainty factors are used in cases where the probabilities arenot known or are too difficult or expensive to obtain. Theevidential reasoning mechanism can manage incrementallyacquired evidence, the conjunction and disjunction ofhypotheses, as well as evidences with different degrees ofbelief. The certainty factors approach also provides betterexplanations of the control flow through a rule-based expertsystem(Negnevitsky, 2002).

This alternative is chosen, because the uncertainty in the proposed expert system design is not tangible, a certainty factor of unity (1.0) is assumed for majority of the hypothesis (conditions) and evidences (actions). Moreover the proposed expert system is not an interactive one; it is automated as soon as an image is selected or acquired. No other user inputs parameters are required in the process of decision making by the system. The question of how certain the expert knowledge is has been settled before rules are designed. The logic in the rules designed is only the path where the expert is certain the rule will follow action given the conditions, otherwise the alternative action will be taken.

The increase of complexity of the system makes uncertainty and vagueness become more tangible (Sikchi and Ali, 2013). It therefore implies that expert systems that are less complex in decision making parameters involve intangible uncertainty issues. It is common in analytical systems whose decision making is based on outcome of experimental tests (such as image analysis or test specimen analysis). Also in expert systems whose decision requires little or no patient related symptom input response.