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Dealing with Uncertainty Reasoning Under Uncertainty Monotonic Reasoning • • • A reasoning process that moves in one direction only. The number of facts in the knowledge base is always increasing. The conclusions derived are valid deductions and they remain so. Reasoning process applied to practical everyday problems must recognize uncertainty • Available information is frequently incomplete • Conditions change over time • There is frequently a need to make an efficient but possibly incorrect guess when reasoning reaches a dead end. Reasoning Under Uncertainty Non-monotonic Reasoning Non-monotonic reasoning (NMR) is based on augmenting absolute truth with beliefs. These tentative beliefs are generally based on default assumptions that are made in light of lack of evidence. A NMR system tracks a set of tentative beliefs and revise those beliefs when knowledge is observed or derived. Reasoning Under Uncertainty • Uncertainty may cause bad treatment in medicine, loss of money in business. • Classic examples of successful expert systems which deal with uncertainty are MYCIN for medical diagnosis and PROSPECTOR for mineral exploration. • In case of medicine, delaying treatment for more tests (for more exact knowledge) may add considerable costs; the patient may die. Reasoning Under Uncertainty Many different types of errors can contribute to uncertainty. 1. data might be missing or unavailable 2. data might be ambiguous or unreliable due to measurement errors 3. the representation of data may be imprecise or inconsistent 4. data may just be user's best guess (random) 5. data may be based on defaults, and defaults may have exceptions Reasoning Under Uncertainty Given these sources of errors, most knowledge base systems incorporate some form of uncertainty management. There are three issues to be considered: 1. How to represent uncertain data. 2. How to combine two or more pieces of uncertain data. 3. How to draw inference using uncertain data Reasoning Under Uncertainty Errors and Induction Deduction is going from general to specific All men are mortal Socrates is a man therefore Socrates is mortal Induction tries to generalize from the specific. My disk has never crashed. Inductive therefore my disk will never crash. Reasoning Under Uncertainty Inductive arguments can never be proven correct (except mathematical induction). Inductive arguments can provide some degree of confidence that the conclusion is correct. Deductive errors or fallacies may also occur If p implies q q is true therefore p Example If the valve is in good condition then the output is normal The output is normal Therefore, the valve is in good condition Uncertainty is major problem in knowledge elicitation, especially when the expert's knowledge must be quantized in rules. Approaches in Dealing with Uncertainty Numerically oriented methods: • • • • Bayes’ Rules Certainty Factors Dempster Shafer Fuzzy Sets Quantitative approaches • Non-monotonic reasoning Symbolic approaches • Cohen’s Theory of Endorsements • Fox’s semantic systems Classical Probability • This is also called a priori probability. It is assumed that all possible events are known and that each event is equally likely to happen (rolling a die). • Prior or unconditional probability is the one before the evidence is received. • Posterior or conditional probability is the one after the evidence is received. Theory of Probability Formal theory of probability can be made using 3 axioms: Axiom 1 0 =< P(E) =< 1 Axiom 2 P( E ) 1 i where Ei, i=< 1=< n are mutually i exclusive P(E) + P(E') = 1 Axiom 3 P( E1 E2 ) P( E1 ) P( E2 ) P( E1 E2 ) Theory of Probability Experimental or Subjective Probabilities • In contrast to the prior approach, experimental probability defines the probability of an event P(E) as the limit of a frequency distribution. P(E) =[ lim (N-> infinity)] f(E)/N This type of probability is called a posterior probability. A subjective probability is a belief or opinion expressed as a probability rather than a probability based on axioms or empirical measurements. This is applied on the decisions for non-repeatable events. Theory of Probability Compound Probabilities • What is the probability of rolling a die with an outcome of even number divisible by 3. Event A= {2, 4, 6} Event B = {3, 6} A B 6 P( A B) n( A B ) 1 n( s ) 6 P( A B) P( A) P( B) Theory of Probability The two events are called stochastically independent events if and only if the above formula is true. Stochastic is a Greek word meaning "guess". It is commonly used as a synonym for probability, random or chance. The probability of rolling a die with an outcome of even number or divisible by 3. P( A B) P( A) P( B) P( A B) = 3/6 + 2/6 – 1/6 = 4/6 Theory of Probability Conditional Probabilities • The probability of an event A, given that event B occurred, is called a conditional probability and indicated by P(A|B). P( A B) P( A | B) forP( B) 0 P( B) P( A | B) P( B) P( A B) Baye's Theorem Baye's Theorem in terms of events E, and hypothesis, P( E H i ) P( H i | E ) P( E H j ) j P( E | H i ) P( H i ) P( E | H j ) P( H j ) j P( E | H i ) P( H i ) P( E ) Baye's Theorem The conditional probability, P(A|B), states the probability of event A given that event B occurred. The inverse problem is to find the inverse probability which states the probability of an earlier event given that a later one occurred. Example: Probability of chosing brand X given it has crashed. This is inverse or posterior probability. Example Table below shows hypothetical disk crashes using a brand X drive within one year X X’ Crash C No Crash C’ 0.6 0.2 0.1 0.1 0.7 0.3 Total of Columns 0.8 0.2 1.0 Total of Rows P(C|X) = ? P(C|X) = P(C X) / P(X) = 0.6 / 0.8 = 0.75 P(C|X’) = P(C X’) / P(X’) = 0.1 / 0.2 = 0.50 P(X|C) = ? P(X|C) = P(C X) / P(C) = 0.6 / 0.7 = 6/7 P(X|C) = P(C|X) P(X) / P(C) = 0.75 * 0.8 / 0.7 = 0.6 / 0.7 Example Suppose, statistics show that Brand X drive crashes with a probability of 75% within one year and non-Brand X drive crash within one year is 50%. The inverse question is, what is the probability of a crashed drive being brand X or non-brand X. Hypothetical Reasoning and Backward Induction • Bayesian decision making is used in PROSPECTOR to decide favorable sites for mineral exploration. • Generally conditional probability is forward in time, while a posterior probability is backward in time. • Example of Bayesian decision making under uncertainty. Oil exploration • If there is no evidence for or against we may guess that P(O) = P(O') = 0.5 • We may believe that the chances are better for finding oil. P(O) = 0.6, P(O') = 0.4 • Assume the probabilities for the outcomes of seismic test for oil exploration as: P(+|O) = 0.8, P(-|O) = 0.2 (false -) P(+|O')= 0.1 (false +), P(-|O') = 0.9 Using above conditional (prior) probabilities we can construct the initial probability tree . . Advantages and Disadvantages of Bayesian Methods • Bayesian methods have support of probability theory and have well defined semantics for decision making. Disadvantages are • They require significant amount of probability data to construct a knowledge base. • If the probabilities are statistical, sample size must be sufficient. If they are provided by an expert then their comprehensiveness and consistency must be queried. • Reducing associations between the hypotheses and the evidences to simple numerical values removes relevant information necessary for reasoning (explanation of how a conclusion is reached). Reasoning with Certainty Factors During the development of MYCIN, researchers developed certainty factors formalism for the following reasons: • The medical data lacks large quantities of data and/or the numerous approximations required by Bayes' theorem. • There is a need to represent medical knowledge and heuristics explicitly, which can not be done when using probabilities. • Physicians reason by capturing evidence that supports or denies a particular hypothesis. Certainty Factor (CF) Formalism Eg of MYCIN rule IF the stain of the organism is gram pos AND the morphology of the organism is coccus AND the growth of the organism is chains THEN there is evidence that the organism is streptococcus CF 0.7 Given the evidence a doctor only partially believe the conclusion • General Form IF E1 And E2 ….THEN H CF = Cfi where E= evidence & H is the conclusion Certainty Factor (CF) Formalism • A measure of belief, MB(h, e) indicates the degree to which our belief in hypothesis, h, is increased based on the presence of evidence, e • A measure of disbelief, MD(h, e), indicates the degree to which our disbelief in hypothesis, h, is increased based on the presence of evidence, e. When p(h | e) = 0 MB(h, e) = 0 MD(h, e) = 1 p(h | e) = 1 MB(h, e) = 1 MD(h, e) = 0 Certainty Factor (CF) Formalism CF interpretation Certainty Factor (CF) Formalism • CF(h | e) = MB(h, e) - MD(h, e) -1 < CF < 1 • When there is total belief – CF = 1, and When there is a total disbelief in hypothesis – CF = -1 When there is no evidence to make judgment – CF = 0 • • -1 F range of disbelief 0 range of belief 1 T Certainty Factor (CF) Formalism • Composite CF can be calculated as follows: CFcomp(h, e) = MBcomp(h, e) - MDcomp(h, e) • For P1 and P2 premises of the rule, CF(P1 and P2)= MIN((CF(P1), CF(P2)) CF(P1 or P2) = MAX ((CF(P1), CF(P2)) Certainty Factor (CF) Formalism For example consider a rule in a knowledge base: • (P1 and P2) or P3 R1(.7) and R2(.3) • If CFs for P1, P2, and P3 are 0.6, 0.4, and 0.2, respectively then R1 and R2 may be anticipated with CFs 0.28 and 0.12 respectively. CF(P1(0.6) and P2(0.4)) = MIN(.6, .4) = 0.4 CF((0.4) or P3(0.2)) = MAX (0.4, 0.2) = 0.4 CF(R1) = .7 * .4 = .28 CF(R2) = .3 * .4 = .12 Certainty Factor (CF) Formalism Two properties that are required of the combination operation are: Commutative – The value should not depend on the order in which the rules are taken. Asymptotic – The more evidence we have for the belief in a conclusion the higher should be the certainty factor, but if it is not absolutely certain, then it should remain below 1. Certainty Factor (CF) Formalism Propagation of Certainty Factors When there are two or more rules supporting the same conclusion CFs are propagated as follows: CFrevised = CFold + CFnew(1 - CFold) if both CFold and CFnew > 0 = CFold + CFnew(1 + CFold) if both CFold and CFnew < 0 = otherwise Certainty Factor Example In a murder trial the defendant is being accused of a first degree murder (hypothesis).The jury must balance the evidences presented by the prosecutor and the defense attorney to decide if the suspect is guilty. RULE001 IFthe defendant's fingerprints are on the weapon, THEN the defendant is guilty. CF=0.75 RULE002 IFthe defendant has a motive, THEN the defendant is guilty. CF=0.60 RULE003 IFthe defendant has a alibi, THEN he is not guilty. CF=-.80 Certainty Factor Example We start with CF = 0.0 for the defendant being guilty. • After submission of the evidence 1 (fingerprints on the weapon) CFcomb1 = CF rule1's conclusin * CF evid1 = 0.75 * 0.90 = 0.675 CF revised = CF old + CF new * (1 - CFold) = 0.0 + 0.675(1-0.0) = 0.675 Example of CFs Propagation CFcon1=CFnew=0.675 CFold=0.0 Guilty CFnew=0.675 Guilty CF = 0.0 CFrevised=0.675 fingerprints on weapon CFevid1=0.90 CFrule1=0.75 CFrevised=CFold + CFnew*(1-CFold) =0.0 + 0.675*(1-0.0) =0.675 RULE 1. IF the defendant’s fingerprints are on the weapon THEN the defendant is guilty CFcon1=CFevid1*CFrule1 (single premise rule) =0.9*0.75 =0.675 Certainty Factor Example The defendant’s mother in law says that he had the motive for slaying CFnew = CFcomb2 = CF rule2's conclusin * CF evid2 = 0.60 * 0.50 = 0.30 CF revised = CF old + CF new * (1 - CFold) = 0.675 + 0.30(1-0.675) = 0.7725 CFcon2=CFnew=0.30 Motive exists CFevid2=0.50 CFrule2=0.60 CFold=0.675 Guilty CFnew=0.30 Guilty CFrevised=0.772 CFrevised=0.675 CFrevised=CFold + CFnew*(1-CFold) =0.675 + 0.30*(1-0.675) =0.7725 RULE 2. IF the defendant has a motive THEN the defendant is guilty of the crime CFcon2=CFevid2*CFrule2 =0.50*0.60 =0.30 (single premise rule) Certainty Factor Example A respected judge witnesses for alibi, so a cf of 0.95 is assigned for this evidence CFcomb3 = CF rule3's conclusin * CF evid3 = 0.95 * (-0.80) = -0.76 CFrevised = = (0.7725 - 0.76) / (1 - 0.76) = 0.052 CFcon3=CFnew=-0.76 Guilty CFrevised=0.772 CFold=0.772 CFnew=-0.76 Guilty CFrevised=0.052 CFold CFnew CFreviced= 1 min(| CFold |,| CFnew | ) Alibi found CFevid3=0.95 CFrule3= -0.80 RULE 3. IF the defendant has an alibi THEN he is not guilty CFcon3=CFevid3*CFrule3 =0.95*(-0.80) = -0.76 = (0.772-0.76)/(1-0.76) = 0.052 Certainty Factor Example Confidence Factor in guilty verdict after introduction of all evidences is: Advantages of Certainty Factors • It is a simple computational model that permits experts to estimate their confidence in conclusions being drawn. • It permits the expression of belief and disbelief in each hypothesis, allowing the expression of the effect of multiple sources of evidence. • It allows knowledge to be captured in a rule representation while allowing the quantification of uncertainty. • The gathering of the CF values is significantly easier than the gathering of values for the other methods. No statistical base is required – you merely have to ask the expert for the values. Difficulties Deep Inference Chains If we have a chain of inference such as: IF ATHEN B CF=0.8 IF B THEN C CF= 0.9 Then because of the multiplication of CFs the resulting CF decreases. For example if CF(A) = 0.8, then CF(C) = .8*.8*.9 = .58 With long chain of inferences the final CF may become very small Difficulties Many Rules with same Conclusion The more rules with the same conclusion the higher the CF value. If there are many rules then CF can become artificially high. Difficulties Conjunctive Rules If a rule has a number of conjunctive premises, overall CF may be reduced too much. IF sky dark AND temperature dropping THEN will rain 0.9 If CF(sky dark) = 1, CF(temperature dropping) = .1 then CF(will rain) = min(1, .1)*.9 = .09 whereas if we had IF the sky dark THEN will rain 0.7 IF temperature dropping THEN will rain 0.5 CF1 = 1 * .7 = 0.7, CF2 = .1 * .5 = 0.05 CF (will rain) = .7 + .05*(1 - .7) = .7 + 0.015 = .715 Fuzzy Logic In everyday speech we use vague or imprecise terms to describe properties. Fuzzy logic was developed by Zadeh to deal with these imprecise values in a mathematical way. Fuzzy Logic • It will allow us to deal with fuzzy rules IF the temperature is cold THEN the motor speed stops IF speed is slow THEN make acceleration high. Fuzzy Sets • In ordinary set theory, an element from the domain is either in a set or not in a set. • In fuzzy sets, a number in the range 0-1 is attached to an element – the degree to which the element belongs to the set. • A value of 1 means the element is definitely in the set • A value of 0 means the element is definitely not in the set • Other values are grades of membership. • Formally a fuzzy set A from X is given by its membership function which has type A : X [0, 1] Fuzzy Sets Fuzzy set of small men Small men – Simpler Curve Fuzzy Sets • The following figure shows the representation of three fuzzy sets for small, medium and tall men. We see that a man of height 4.8 feet is considered both small and medium to some degree. Boolean Operations The Boolean operations of union, intersection, and complement can be defined in the straightforward manner. Complement The operation is A (x) = 1 - A (x) Boolean Operations Intersection The intersection of two fuzzy sets A and B is given by AB (x) = min({A (x), B (x)}) Union The union of two fuzzy sets A and B is given by AB (x) = max({A (x), B (x)}) Fuzzy Reasoning In this section, fuzzy rules and how inference is performed on these rules is presented. This will be illustrated by a fuzzy system used to control an air conditioner. The variables to be used (with fuzzy values) are temperature (of the room) and speed (of the fan motor). Fuzzy Reasoning The rules are given as follows: • IF the temperature is cold THEN motor speed stops • IF the temperature is cool THEN motor speed slows Temperature Fuzzy Sets • IF the temperature is just right THEN motor speed medium • IF the temperature is warm THEN motor speed fast • IF the temperature is hot THEN motor speed blast Speed Fuzzy Sets Fuzzy Reasoning • In a fuzzy system all the rules fire in parallel, although in the end many will not contribute to the output. • What we need to determine, in the above system is, given a particular value of the temperature how do we calculate the motor speed. Fuzzy Reasoning • Now, the temperature can be measured fairly accurately, but it will lie in several fuzzy sets. For example if the temperature were 17C then from the figure we see that it is about 25% cool and 80% just right. Fuzzy Reasoning • This means that rules 2 and 3 will contribute to the output speed of the motor. • The fuzzy set for the output can be calculated by multiplying the slow graph by .25 and the medium graph by .80 assuming the contribution is proportional to the fuzzy values of the input temperature Fuzzy Reasoning • One way to amalgamate two sets is to sum the values (with a maximum of 1). Amalgamated sets and average Fuzzy Reasoning • Other ways of amalgamation (e.g. taking maximum) are possible. • Now we need to determine the actual speed of the motor. This can be done by finding the average value of the curve – I.e. the position where the areas on either side of the perpendicular through this point are equal. acknowledgement • Phil Grant: University of Wales Swansea