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Brics “ Uncertainty comes in many forms and in real world problems it is usually impossible to avoid uncertainties. Further looking towards the design of intelligent systems, one of the common problems is “ how to handle uncertain, imprecise, and incomplete information ?” So that resulting designed system come closer to the human intelligence level.” Handling Uncertainties - using Probability Theory to Possibility Theory By Ashutosh Dwivedi, D Mishra and Prem k Kalra 1. Introduction Uncertainty comes in many forms and in the real world problems it is usually impossible to avoid uncertainties. At the empirical level, uncertainties are associated with the measurement errors and resolution limits of measuring instruments. At the cognitive level, it emerges from the vagueness and ambiguities in natural language. At the social level, it is created and maintained by people for different purposes (e.g. privacy, secrecy etc.). The processing of uncertainties in the information’s have always been a hot topic of research since mainly the 18th century. Up to middle of the 20th century, most theoretical advances were devoted to the theory of probabilities through the works of eminent mathematicians like J. Bernoulli (1713), A. De Moivre (1718), T. Bayes (1763), P. Laplace (1774), K. Gauss (1823), S. Poisson (1837), E. Borel (1909), R. Fisher (1930), A. Kolmogorov (1933), B. De Finetti (1958), L. Savage (1967), T. Fine (1973), E. Jaynes (1995), to name just few of them. The second half of the 20th century had become very prolific for the development of new original theories dealing with uncertainties and imprecise information’s. Mainly three major theories are available now as alternative of the theory of probability for the automatic plausible reasoning in expert systems: the fuzzy set theory developed by L. Zadeh in sixties (1965), the Shafer’s theory of evidence in seventies (1976) and the theory of possibilities by D. Dubois and H. Prade in eighties (1985). 2. Sources of Uncertainties and Inexact Information The sources of uncertainty can be given as follows: 2.1. Incomplete Domain Knowledge The knowledge about the domain in which work is done may be imprecise or incomplete. Incomplete domain knowledge compels to use some form of rules or heuristics, which may not always give appropriate or correct results as outcome of the work. 2.2. Noisy and Conflicting Data Any data when it is collected with some purpose will always not be accurate and complete. Evidences from different sources may be conflicting or noisy. For example it can be shown that the statement “Ram is tall” is vague and results in sources of uncertainty. Suppose given a number x represents the height of Ram in feets then x, xÎ [0,8] is given. It is to be decided if this number is large enough to decide the tallness of a person or not. One way to answer this question is that if x ³ d , (where d is some threshold or limit) then it is large enough to say that a person is tall else he is not. Second way can be thought as to mark either agree or disagree or place no mark based on subjective knowledge of source and this arises the uncertainty whether is large. Another way is to create a scale of measurement. If we are on one end of the scale then we completely disagree and if we are on another extreme end then we completely agree. But if we are in between, we have some value associated with our measurement. Thus there is a lack of sharp boundaries for the result. 2.3 Incomplete Information It occurs when a factor in a decision or model 1 is not known. Sometimes it can be resolved (through research, enquiry etc.) but not always. Some factors are necessarily uncertain because they are indeterminate, e.g. all future developments. Sometimes they may be determinate but not practically measurable. Factors or parameters change with time, hence it also produces a kind of incomplete information which further leads to uncertainty. 2.4 Selection among the alternatives When we have a situation where there are several alternatives available, there may be disagreement in a choosing among several alternatives. This procedure also leads to uncertainty. Arising from interpretation of data by a person. (B) Linguistic Uncertainty (i) Vagueness (ii) Ambiguity (iii) Context Dependency (Iv) Nonspecificity Human reasoning often involves inexact information. The sources and the nature of inexact information may be different for different problem domain. For example the possible logical sources of the inexactness of information for the medical domain are: Lack of adequate data Inconsistency of data · Inherent human fuzzy concept · Matching of similar situations · Different opinions Ignorance · · Lack of available theory to describe a particular situation · · 2.5 Linguistic Imprecision In real world people share their thoughts by speaking. People often make use of imprecise terms to express their own subjective knowledge. But when such imprecise terms are interpreted by others, who are not familiar with the meaning or their context changes, this situations may lead to uncertainty. There are mainly two types of uncertainty as proposed by Regan (2002). The first one is called as Epistemic Uncertainty (indeterminate facts) and other one is Linguistic Uncertainty (indeterminacy in language). (A) Epistemic Uncertainty Following factors can be considered the causes of such uncertainties. (i) Systematic error It is the result of bias or miscalibrationc in experiment while doing some experimentation. (ii)Measurement error It is nothing but imprecision or inaccuracy in the measurement procedure itself. (iii) Natural Variation This is due to the unpredictable changes that occur in surroundings. (iv) Inherent Randomness This is due to the factors contributing (i), (ii) or (iii) or combination of them. (v) Imperfect Data The data or information used in decision making procedure may be corrupt and imperfect which further leads to wrong decision. (vi)Subjective Judgment 2 For example, at the starting of the consultation session, a doctor asks the patient and the person accompanying him/her about the condition of the patient. When the patient describes his/her situation, there are inexactness of informations such as source of disease, period of suffering or about symptoms of disease and the doctor has to deal with this kind of incomplete, inexact information’s. Thus the proper management of inexact information is necessary for decision making. 3. Methods to Handle Uncertainties Different methods for handling uncertainties are as follows: 1. Probability Theory 2.Certainty Factor 3.Dempster Shafer Theory of Evidence 4.Possibility Theory 3.1 Probability Theory Probability has been discussed for a long time. Indeed, the arguments go back over two or three centuries. There are three schools of thought, each leading to approximately similar mathematical methods. Three views of probability: • Frequentist • Mathematical • Bayesian (knowledge-based) One school of thought treats the word "Probability" as a synonym for "frequency" or "fraction" of events. If we are trying to determine the fraction of defects in a lot (an enumerative problem), we imagine that the lot before us has been drawn at random for a "super lot", that is, a lot of lots. Then we sample it and based on the sample, attempt to say something useful about the entire lot and, by inference, the characteristics of the "super lot". 2 Other people like the "long run frequency" interpretation. The idea is that if we were to repeat what we are doing, over and over again, if only in our head, we can imagine that the fraction of defects (or whatever we are finding) is in some way, the "probability". There are the "equal a-priori" thinkers, who, like Laplace, considered that every problem could be considered broken into cases, which had "equal a-priori probability". Tossing of dice or dealing of cards are some examples. Then there are the "subjectivists" who are Bayesians, but they start from a different basis. They say that the probability measures strength of belief. To them a probability assignment represents a logical combination of what we believe and what the data tell us to believe. As a result, the end up using Bayes equation and, except for the way they get started on a problem. The math is similar but, again, there are important differences in the way they would formulate the problem and how they take into account fact, what is known and what is unknown. This debate continues to occur among the Bayesian community. Probability represents a unique encoding of incomplete information. The essential task of probability theory is to provide methods for translating incomplete information into this code. The code is unique because it provides a method, which satisfies the following set of properties for any system: (a) If a problem can be solved in more than one way, all ways must lead to the same answer. (b) The question posed and the way the answer is found must be totally transparent. There must be no "laps of faith" required to understand how the answer followed from the given information’s. (c) The methods of solution must not be "adhoc". They must be general and admit to being used for any problem, not just a limited class of problems. Moreover the applications must be totally honest. For example, it is not proper for someone to present incomplete information, develop an answer and then prove that the answer is incorrect because in the light of additional information a different answer is obtained. (d)The process should not introduce information that is not present in the original statement of the problem. 3.1.1. Basic terminologies used in Probability Theory and Uncertainty handling using probability theory (through example): (a) The Axioms of Probability The language system of probability theory, as most other systems, contains a set of axioms (un-provable statements which are known to be true in all cases) that are used to constrain the probabilities assigned to events. Three axioms of probability are as follows: 1. All values of probabilities are between zero and one i.e. A[0 P ( A) 1] 2. Probabilities of an event that are necessarily true have a value of one, and those that are necessarily false have a value of zero i.e. P (True ) 1 and P ( False) 0 . 3. The probability of a disjunction is given by: A, B[ P ( A B ) P ( A) P ( B ) P ( A B ) An important result of these axioms is calculating the negation of a probability of an event . If we know that P ( A) 0.55 then it follows from these axioms that, P ( A) 1 P ( A) 0.45 . (b) The Joint Probability Distribution The joint probability distribution is a function that specifies a probability of certain state of the domain given the states of each variable in the domain. Suppose that our domain consists 3 Of three random variables or atomic events {x1 , x2 , x3 } . Then an example of the joint probability distribution of this domain where x1 = 1, x2 0 and x3 = 1 would be P( x1 , x2 , x3 ) = some value depending upon the values of different probabilities of x1 , x2 and x3 . The joint probability distribution will be one of the many distributions that can be calculated using a probabilistic reasoning system. (c) Conditional Probability and Bayes' Rule Suppose a rational agent begins to perceive data from its world, it stores this data as evidence. This evidence is used to calculate a posterior or conditional probability which will be more accurate than the probability of an atomic event without this evidence (known as a prior or unconditional probability). Example: -The probability that a car can start is 0.96 . -The probability that a car can move is 0.94. -The probability that a car both starts and moves is 0.91. Now suppose that the agent perceives information that the car starts. The agent can now calculate a more accurate probability that the car moves given that the car started. Ø What is the probability that the car moves given that the car starts? In order to calculate the example above, we need a mathematical formula to relate the conditional probability to the probability of the given atomic event, and the probability of the conjunction of the two events occurring. Lets define: P( A | B ) = Probability of happening of A where B has already happened. P( A Ç B) =Probability of happening of A & B at same time. P( B) = Probability of happening of B alone. So the relation P (A / B) = P( A Ç B) P(B) (1) Now using this equation for conditional probability, to calculate our example: 4 P ( moves | starts ) = P ( moves Ç starts ) P(starts ) 0.91 = = 0.95(approx ) 0.96 The agent would be able to update the prior probability of the car moving from 0.94 to 0.95 after perceiving the evidence that the car started. Algebraic manipulation of the equation for conditional probability yields the product rule. P ( A Ç B) = P( A | B).P( B) And , since P( A | B) = P ( A Ç B) = P( B Ç A) P( B | A).P( A) P( B) Using the equations for the product rule above, and since P( A Ç B) = P( B Ç A) , we can now derive Baye’s Rule as: (2) P( B | A).P( A) P( A | B) = P( B) Baye’s Rule is of great importance in practice. Using the example above again, let us add the knowledge that the probability that the car moves given that the car starts is 0.95. We can now find the probability that the car starts given that the car moves. P(moves | starts ).P( starts ) P(moves ) 0.95 ´ 0.96 = = 0.97(approx) 0.94 P( starts | moves ) = (D) Conditional Independence When the result of one atomic event, A, does not affect the result of another atomic event, B, those two atomic events are know to be independent of each other and this helps to resolve the uncertainty. This relationship has the following mathematical property : P( A, B) = P( A).P( B) Another mathematical implication is that P( A | B) = P( A) Independence can be extended to explain irrelevant data in conditional relationships. In previous example, assume that we want to know the probability of a car starting. We are given information that the car's spark plugs fire, and that the battery is charged, P (starts|sparkplugs Ç battery). therefore irrelevant to finding the probability that the car starts. P (starts | sparkplugs Ç battery ) = P( starts | sparkplugs ) Advanta ges and Pr oblems with Probabilistic Method: Advantages (1) Formal foundation (2) Reflection of reality (a posteriori) (3)Well-defined semantics for decision making Problems (1) This may be inappropriate. • The future is not always similar to the past. (2) Complex data handling • Requires large amounts of probability data. • Non-sufficient sample sizes. (3) Inexact or incorrect •Especially for subjective probabilities. •Subjective evidence may not be reliable. (4) Ignorance •Probabilities must be assigned even if no information is available. •Assigns an equal amount of probability to all such items. •Independence of evidences assumption often not valid. (5) Non-local reasoning •Requires the consideration of all available evidence, not only from the rules currently under consideration. •High computational overhead. (6) No compositionality •Complex statements with conditional dependencies cannot be decomposed into independent parts. •Relationship between hypothesis and evidence is reduced to a number. (7)Always requires prior probabilities which are very hard to found out apriori. 3.2. Certainty Factor Certainty factor is used to express how accurate, truthful or reliable you judge a predicate to be. This issue is how to combine various judgments. It is neither a probability nor a truth value but a number that reflects the overall belief in a hypothesis given available information. It may apply both to facts and to rules, or rather to the conclusion(s) of rules. They are logic and rules which provide all or nothing answers. Decision is based on evidences and interpretation of evidences is subjective. Idea here is to minimize amount of computation required to update the probability, as evidences are accumulated incrementally. Approach used is to model human expert belief about events rather than probabilities of events. Probability described frequency of occurrence of reproducible events but in CF we represent amount of support for a particular fact based on some other facts. Probability does not account for positive and negative examples but CF does. CF (Certainty factor) denotes the belief in a hypothesis H given that some pieces of evidence E are observed. No statement about the belief means that no evidence i.e. separation of belief, disbelief, ignorance is possible. It ranges from -1 to +1 (Scale on same range) where CF=-1 no belief at all in the hypothesis (H) CF= +1 fully believes the hypothesis (H) CF=0 nothing is known about the hypothesis (H) Certainty factor combine belief and disbelief into a single number based on some evidence. Strength of the belief or disbelief in H depends on the kind of the observed evidence E: MB( H , E ) - MD( H , E ) CF ( H , E ) = 1 - min[ MB( H , E ) - MD( H , E )] MB is a measure of belief, MD is a measure of disbelief. Suppose MB( E ® H ) = 0.7 and MB( E ® H ' ) = -0.3. In probability theory this is true but not in CF. • Positive CF implies evidence support hypothesis since MB < MD • CF of 1 means evidence definitely supports the hypothesis. • CF of 0 means either there is no evidence or that the belief is cancelled out by the disbelief. • Negative CF implies that the evidence favors negation of hypothesis since MB < MD 5 Certainty Factor Algebra : There are rules to combine CFs of several facts: Interpretation of CF value: Uncertain terms Definitely Not Almost Not Probably Not May be not Unknown May be Probably Almost Certain Definitely (CF ( x1 ) AND CF ( x2 ) = min(CF ( x1 ), CF ( x2 )) (CF ( x1 ) OR CF ( x2 ) = max(CF ( x1 ), CF ( x2 )) A rule may also have a certainty factor CF (rule) . CF (action) = CF (condition) * CF (rule) Consequent of multiple rules: Suppose the following IF A is X THEN C is Z IF B is Y THEN C is Z What certainty should be attached to C having Z if both rules are fired? CFR1 (C ) + CFR2 (C ) - CFR1 (C )* CFR2 (C ) When CFR (C ) and CFR2 ( C ) are both 1 positive CFR1 (C) CFR2 (C) CFR1 (C)*CFR2 (C) When CFR1 (C) and CFR2 (C) are both negative CFR1 (C) + CFR2 (C) 1- min[| CFR1 (C) |,| CFR2 (C) |] When CFR1 (C ) and CFR2 (C ) are of opposite sign Where, CFR (C ) is the confidence in hypothesis established by Rule 1 and CFR (C ) is the confidence in hypothesis established by Rule 2. CF -1 -0.8 -0.6 -0.4 -0.2 or 0.0 0.4 0.6 0.8 1 Difficulties with certainty Factors •Results may depend on the order in which evidences are considered in some cases. For example: Suppose CF ( A) = 0.4, CF ( B1 ) = 0.3 and CF ( B2 ) = 0.6 ( A Ç ( B1 È B2 ) is C If then it is evaluated as CF (C )= min(0.4,max(0.3,0.6)) = min(0.4,0.6) = 0.4 If they are evaluated as follows : ( A Ç B1 then CF(C(1))) and if ( A Ç B2 then CF(C(1))) Then CF(C) = 0.3 + 0.4 - 0.3 ´ 0.4 = 0.58 Thus, the results vary with the way in which the evidences are considered. • Combination of non-independent evidences is unsatisfactory. • New knowledge may require changes in the certainty factors of existing knowledge. • Not suitable for long increase chains. Comparison of Certainty Factors with Probability Theory 1 2 Example CF(Sheep is a dog) = 0.7; CF(Sheep has wings) = -0.5 CF(Sheep is a dog and has wings) = min(0.7, 0.5) = -0.5 Suppose there is a rule. If x has wings then x is a bird. Let the CF of this rule be 0.8 If (Sheep has wings) then (Sheep is a bird) = -0.5 * 0.8 = -0.4 6 •Probability theory is used in areas where statistical data is available whereas certainty factors are used in the areas where expert’s judgments are used. e.g., Diagnosis of disease. •Assumptions of conditional independence cannot be made. •In probability theory Measure of Beliefs: (MB(Evidence->Hypothesis + MB (Evidence> Not hypothesis) = 1. But in certainty factor this is not true. •Sometimes when two events A and B are given then CF(A B) = CF(A B) . •Used in cases where probabilities are not known or too difficult or expansive to obtain. 3.3. Dempster-Shafer Theory of Evidence Dempster-Shafer Theory (DST) is a mathematical theory of evidence. The seminal work on the subject is done by Shafer [Shafer, 1976], which is an expansion of previous work done by Dempster [Dempster, 1967]. In a finite discrete space, Dempster-Shafer theory can be interpreted as a generalization of probability theory where probabilities are assigned to sets as opposed to mutually exclusive singletons. In traditional probability theory, evidence is associated with only one possible event. In DST, evidence can be associated with multiple possible events, e.g., sets of events. DST allows the direct representation of uncertainty. There are three important functions in DempsterShafer theory: • the basic probability assignment function (bpa or m), • the Belief function (Bel), and the • Plausibility function (Pl). Basic probability assignment does not refer to probability in the classical sense. The bpa, represented by m, defines a mapping of the power set to the interval between 0 and 1. m: P(X) 0,1] (1) The bpa of the null set is 0 i.e. m (Æ ) = 0 (2) Where Æ is the null set The value of the bpa for a given set A (represented as m(A), A is a set in the power set AÎ P(X)), expresses the proportion of all relevant and available evidence that supports the claim that a particular element of X (the universal set) belongs to the set A but to no particular subset of A. m (A) = 1 (3) AÎP (X ) The value of m(A) pertains only to the set A and makes no additional claims about any subsets of A. Any further evidence on the subsets of A would be represented by another bpa, i.e. B A, m(B) would the bpa for the subset B. The summation of the bpa’s of all the subsets of the power set is 1. As such, the bpa cannot be equated with a classical probability in general. From the basic probability assignment, the upper and lower bounds of an interval can be defined. This interval contains the precise probability of a set of interest (in the classical sense) and is bounded by two nonadditive continuous measures called Belief and Plausibility. The lower bound Belief for a set A is defined as the sum of all the basic probability assignments of the proper subsets (B) of the set of interest (A) (B - A). Bel(A) = m(B) (4) BÍA The upper bound, Plausibility , is the sum of all the basic probability assignments of the sets (B) that intersect the set of interest (A)(B Ç A ¹ Æ ). Pl(A) = m(B) (5) BÇA ¹ Æ The two measures, Belief and Plausibility are nonadditive. It is possible to obtain the basic probability assignment from the Belief measure with the following inverse function: m(A) = (-1) |A-B| (6) Bel(B) BÍA where|A-B| is the difference of the cardinality of the two sets. In addition to deriving these measures from the basic probability assignment (m), these two measures can be derived from each other. For example, Plausibility can be derived from Belief in the following way: Pl(A) = 1- Bel (not A) and Bel (A) = 1 - Pl(not A) (7) where not A is the classical complement of A. This definition of Plausibility in terms of belief comes from the fact that all basic assignments must sum to 1. And because of this, given any one of these measures (m(A), Bel(A), Pl(A)) it is possible to derive the values of the other two measures. The precise probability of an event (in the classical sense) lies within the lower and upper bounds of Belief and Plausibility, respectively. Bel(A)(10) < P(A) >Pl(A) The probability is uniquely determined if Bel (A) = Pl(A). In this case, which corresponds to classical probability, all the probabilities, P(A) are uniquely determined for all subsets A of 7 the universal set X. Otherwise, Bel (A) and Pl(A) may be viewed as lower and upper bounds on probabilities, respectively, where the actual probability is contained in the interval described by the bounds. Dempster’s rule, we find that m(brain tumor) = Bel(brain tumor) = 1. Clearly, this rule of combination yields a result that implies complete support for a diagnosis that both physicians considered to be very unlikely. RULES FOR COMBINING EVIDENCES Example Combination rules are the special types of aggregation methods for data obtained from multiple sources. These multiple sources provide different assessments for the same frame of discernment and Dempster-Shafer theory is based on the assumption that these sources are independent. Dempster’s rule combines multiple belief functions through their basic probability assignments (m). The combination (called the joint m12) is calculated from the aggregation of two bpa’s m1 and m2 in the following manner: M12(A)= m1(B) m2(C) /( 1- K ) (11) Suppose two experts are consulted regarding a system failure. The failure could be caused by Component A, Component B or Component C. The distributions can be represented by the following: Expert 1: m1(A) = 0.99 (failure due to Component A) m1(B) = 0.01 (failure due to Component B) Expert 2: m2(B) = 0.01 (failure due to Component B) m2(C) = 0.99 (failure due to Component C) B ÇC = A when A ¹ Æ and m12(Æ ) = 0 Expert 1 ; when A= where K = m1(B) m2 (C) Æ (12) A B C Failure cause (13) 0.99 0.01 0.0 m1 Æ K represents basic probability mass associated with conflict. This is determined by the summing of the products of the bpa’s of all sets where the intersection is null. The denominator in Dempster’s rule, 1-K, is a normalization factor. This has the effect of completely ignoring conflict and attributing any probability mass associated with conflict to the null set. The problem with conflicting evidence and Dempster’s rule was originally pointed out by Lotfi Zadeh in his review of Shafer’s book, A Mathematical Theory of Evidence. Zadeh had given a compelling example of erroneous result found by using Dempster Shafer theory. Suppose a patient is seen by two physicians regarding the patient’s neurological symptoms. The first doctor believes that the patient has either meningitis with a probability of 0.99 or a brain tumor, with a probability of 0.01. The second physician believes the patient actually suffers from a concussion with a probability of 0.99 but admits the possibility of a brain tumor with a probability of 0.01. Using the values to calculate the m(brain tumor) with 8 Failure E cause m2 X P A Æ Æ E m1(A)m2(A) m1(B)m2(A) m1(C)m2(A) R A 0.0 0.99 x 0.0 0.01 x 0.0 0.0 x 0.0 T = 0.0 = 0.0 = 0.0 2 Æ B Æ m1(A)m2(B) m1(B)m2(B) m1(C)m2(B) B 0.01 0.99 x 0.01 0.01 x 0.01 0.0 x 0.01 = 0.0 = 0.0099 = 0.0001 C Æ Æ m1(A)m2(C) m1(B)m2(C) m1(C)m2(C) C 0.9 9 0.99 x 0.99 0.01 x 0.99 0.0 x 0.99 = 0.0 = 0.0099 =0.9801 Table 1 : Dempster Combination of Expert 1 and Expert 2 Ø Steps: Using Equations 11-13: 1. To calculate the combined basic probability assignment for a particular cell, simply multiply the masses from the associated column and row. 2. Where the intersection is nonempty, the masses for a particular set from each source are multiplied, e.g., m12(B) = (0.01)(0.01) = 0.0001. 3. Where the intersection is empty, this represents conflicting evidence. For the empty intersection of the two sets A and C associate with Expert 1 and 2, respectively, there is a mass associated with it. m1(A) m2(C)=(0.99)(0.99) =(0.9801). 4. Then sum the masses for all sets and the conflict. 5. The only nonzero value is for the combination of B, m12(B) = 0.0001. 6. For K, there are three cells that contribute to conflict represented by empty intersections. Using Equation 13, K = (0.99)(0.01) + (0.99)(0.01) + (0.99)(0.99) = 0.9999 7. Using Equation 11, calculate the joint, m1(B) m2(B) = (.01)(.01) / [1-0.9999] = 1 Though there is highly conflicting evidence, the basic probability assignment for the failure of Component B is 1, which corresponds to a Bel (B) = 1. This is the result of normalizing the masses to exclude those associated with conflict. This points to the inconsistency when Dempster’s rule is used in the circumstances of significant relevant conflict that was pointed out by Zadeh. CONTEXT RELAVENCE CONFLICT EVIDENCE Type of evidence Amount of evidence Accuracy of evidence SOURCES -Type of sources -Number of sources -Reliability of sources -Dependence between sources -Conflict between sources COMBINATION OPERATION -Type of operation -Algebraic properties -Handling of conflict -Advantages - Disadvantages Figure 1: Important Issues in the Combination of Evidence Much of the research in the combination rules in Dempster-Shafer theory is devoted to advancing a more accurate mathematical representation of conflict. In Figure 1, all of the contextual considerations like the type, amount, and accuracy of evidence as well as the type and reliability of sources and their interdependencies can be interpreted as features of conflict assessment. Once values are established for degree of conflict, the most important consideration is the relevance of the existing conflict. Though conflict may be present, it may not always be contextually relevant. Advantages and Problems of DempsterShafer Theory of Evidence: Advantages Clear, rigorous foundation Ability to express confidence through intervals Proper treatment of ignorance Problems Non-intuitive deter mination of mass probability Very high computational overhead May produce counterintuitive results due to normalization 3.4. Possibility Theory Possibility theory was introduced to allow a reasoning to be carried out on imprecise or vague knowledge, making it possible to deal with uncertainties on this knowledge. Possibility is normally associated with some fuzziness, either in the background knowledge on which possibility is based, or in the set for which possibility is asserted. This constitutes a method of formalizing non-probabilistic uncertainties on events: i.e. a mean of assessing to what extent the occurrence of an event is possible and to what extent we are certain of its occurrence, without knowing the evaluation of the possibility of its occurrence. For example, imprecise information such as “Ram’s height is above 170 cm” implies that any height h above 170 is possible and any height equal to or below 170 is impossible for him. This can be represented by a “possibility” measure defined on the height domain whose value is 0 if h < 170 and 1 if h ³170 (0 = impossible and 1 = possible). When the predicate is vague like in “Ram is tall”, the possibility can be accommodate degrees, the largest the degree, the largest the possibility. Possibility theory was initialized by Zadeh, as a complement to probability theory to deal with 9 uncertainty. Since Zadeh, there has been much research on possibility theory. There are two approaches to possibility theory: one, proposed by Zadeh, was to introduce possibility theory as an extension of fuzzy set theory; the other, taken by Klir and Folger and others, was to introduce the possibility theory in the frame work of Dempster-Shafer’s Theory of evidence. The first approach is intuitively plausible and closely related to the original motivation for introducing possibility theoryrepresenting the meaning of information. The second approach puts possibility theory on an axiomatic basis. 1. The intuitive Approach to Possibility (as proposed by Zadeh in light of Fuzzy set theory): Let F be a fuzzy set in a universe of discourse U which is characterized by its membership function m F (U) , which is interpreted as the compatibility of u U with the concept labeled F. Let X be a variable taking values in U and F act as a fuzzy restriction, R(X), associated with X. Then the proposition "X is F ", which translates into R(X) = F associates a possibility distribution, p X x , with X which is postulated to be equal to R(X). The possibility distribution function, X (u ) ,characterizing the possibility distribution p X is defined to be numerically equal to the membership function F (u ) of F, that is, p X := m F ; Where the symbol := stands for "denotes" or "is defined to be". Example: Let U be the universe of integers and F be the fuzzy set of small integers defined by F = {(1, 1), (2, 1), (3, 0.8), (4, 0.6), (5, 0.4), (6, 0.2)} Then the proposition "X is a small integer" associates with X the possibility distribution p x = F ; in which a term such as (3, 0.8) signifies that the possibility that x is 3, given that x is a small integer, is 0.8. 2. The Axiomatic Approach to Possibility: For consonant body of evidences, the belief measure becomes necessity measure and plausibility measure becomes possibility measure. Hence Bel ( A Ç B) = min[bel ( A), bel(B)] 10 becomes h ( A Ç B ) = min[h ( A), h (B)] And Pl ( A È B) = max[ Pl ( A), Pl(B)] Becomes p ( A È B) = max[p ( A), p (B)] And also ( A) 1 ( A ) ; ( A) 1 ( A) Now if possibility distribution function is defined as: r : U [0,1] This mapping is related to possibility measure as: ( A) max r(X) X A Now the basic probability assignment is defined for consonant body of evidences as: m=(1 , 2 , 3 ,......., n ) n mi = 1 Where å i= and if possibility distribution function is defined as: r ( 1 , 2 , 3 , ............., n ) 1 n Where ri = n å m = åm (A ) k k =i k k =i Example Let three sets represents different geometrical shapes of a figure as A = { circles} ; B = {Octagons} ; C = {Pentagons} There are two expert opinions m1 and m2 regarding the shape of figure. The focal elements are given as follows in table below. Focal elements Null C B BÈC A AÈ C AÈ B A È BÈ C A B C m1 m2 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0.2 0.3 0 0 0 0.5 0 0.1 0.1 0 0.1 0 0.3 0.4 For Expert 1 opinion, we have B Ì (BÈ C)Ì (AÈ B È C) Thus the probability distribution is given by r = ( 1, 1, 0.8, 0.5, 0.5, 0.5, 0.5) But for Expert 2 the evidences are not nested and hence plausibility measures leads to conflict. Comparison Between Possibility Theory and Probability Theory 1. Probability theory is based on measure of one type: Probability measure. Possibility theory is based on the measures of two types : Possibility and Necessity measure. 2. Probability theory: Here body of evidences consists of singletons. Possibility theory: Body of evidences consists of a family of nested subsets. 3. Probability theory: Additivity i.e. P ( A È B) = P( A) + P( B ) - P( A Ç B) Possibility theory: Max/Min Rules i.e. Poss ( A È B) = max[ Poss ( A), Poss ( B )] 4. Probability theory: P( A) + P( A) = 1 Possibility theory: Poss ( A) + Poss ( A) ³ 1 ; Nec( A) + Nec( A) £ 1 max[ Poss ( A), Poss ( A)] = 1 min[ Nec( A), Nec( A)] = 0 5. Probability theory: P ( A) = p ( x) r : X ® [ 0,1] , Poss (A) = max(r) A xÎ A Difference between Probability Measures Possibility Measures and 1. Notion of noninteractivness on possibility theory is analogous to the notion of independence in probability theory. If two random variables x and y are independent, their joint probability distribution is the product of their individual distributions. Similarly if two linguistic variables are non interactive, their joint probabilities are formed by combining their individual possibility distribution through a fuzzy conjunction operator. 2 Even through possibility and probability are different, they are related in a way. If it’s impossible for a variable x to take a value xi , it is also impossible for x to have the value xi , i.e. p X ( xi ) = 0 Þ PX ( xi ) = 0 In general, possibility can serve as an upper hand on probability. Probability Vs Possibility through an Example x ÎA Possibility theory: for Similarity between Probability Measures Possibility Measures and 1. Each model based on a measure that maps the subsets of the frame of discernment W to the [0,1] interval and is monotype for inclusion. The main property of Probability measure is W P :2 ® [0,1] And P ( A È B) = P( A) + P ( B) - P( A Ç B) "A,B Í W Similarly the possibility measure can be given as p :2 W® [0,1] and p ( A È B) = max[p ( A), p ( B)], "A, B Í W 2. Possibility measures degree of ease for a variable to be taken a value whereas probability measures the likelihood for a variable to take a value. 3. Being a measure of occurrence, the probability distribution of a variable must add to exactly 1. Whereas there is no such restriction in possibility distribution since the variable can have multiple possible values. 4. Possibility measures replace the additivity axiom of probability with the weaker subadditivity condition. Consider the number of eggs X that Hans is going to order tomorrow morning. Let p ( u) The degree of ease with which Hans can eat eggs. Let p(u) is the probability that Hans will eat u eggs at breakfast tomorrow. Given our knowledge, assume the value of p ( u) and p(u) as shown in table below: u 1 2 p ( u) 1 1 p(u) 0.1 0.8 3 4 5 6 7 8 1 0.1 1 0 0.8 0 0.6 0 0.4 0 0.2 0 We observe that whereas the possibility that Hans may eat 3 eggs for breakfast is 1, the probability that he may do might quite small e.g. 0.1. Thus, a high degree of possibility does not imply a high degree of probability, nor does a low degree probability imply a low degree of possibility. But if an event is impossible, it is bound to be impossible. Concluding Remark The various forms of uncertainties may be encountered simultaneously and it is necessary that we be able to integrate them. E.g., in common –sense reasoning, two or tree forms of uncertainties are often encountered in the same statement. Here in integration process of various methods of dealing with uncertainties, 11 care must be taken to define the domain of each method. Understanding of the meaning of statements and their translation into appropriate models is delicate. For example, how do we translate “usually bald men are old”. Which of P(bald|old) or P(old|bald) is somehow large? “When x shaves himself, usually x does not die”, which conditioning is appropriate: Pl(dead|shaving) or Pl(shaving|dead)? Is it a problem of plausibility or possibility? These examples are just illustrative of the kind of problems that must be addressed. Every method has some limitations in comparison to other method and no one give the full power to tackle all kind of uncertainties that we face in real life. Almost no work has been done in the area to integrate all methods. References 1.George J. Kir and Bo Yuan (2002), “Fuzzy Sets and Fuzzy Logic”, PHI Publication. 2. Timothy J. Ross (1995), “Fuzzy Logic with Engineering Applications”, McGraw Hill International Applications. 3.Philippe Smets (1999), “Verities of ignorance and the need for well-founded theories”, IRIDIA University, Belgium 4. Philippe Smets (1999), “Theories of Uncertainties”, IRIDIA University, Belgium 5. Eric Raufaste , Rui da Silva Neves and Claudette Marine`, “Testing the Descriptive validity in human judgments of uncertainty”, Reprint Version to appear in “Artificial Intelligence”. 6. Helen M. Regan, Mark Colyvan and Mark A. Burgman (1999), A proposal for fuzzy International Union for the Conservation of Nature (IUCN categories and criteria“, Biological Coservation, Elsevier Science Ltd. 7. From class notes and exercises of course “Fuzzy Sets, Logics, Systems and Applications EE658 (2004)” taught by Prof. P K Kalra at IIT Kanpur. About the author: Ashutosh Dwivedi is pursuing his Ph. D. in the Department of Electrical Engineering in Indian Institute of Technology Kanpur, India. He obtained Masters of Technology in Control System from MBM Engineering College, Jodhpur. His major areas of study include Control System, Neural Networks and Fuzzy Logic. About the author: Deepak Mishra is pursuing his Ph. D. in the Department of Electrical Engineering in Indian Institute of Technology Kanpur, India. He obtained Masters of Technology in Instrumentation from Devi Ahilya University, Indore in 2003. His major field of study is Neural Networks and Computational Neuroscience. (Home page: http://home.iitk.ac.in/~dkmishra) About the author: Professor Prem Kumar Kalra obtained his PhD in Electrical Engineering from University of Manitiba, Canada. He is currently Professor and Head of Electrical Engineering Department at IIT Kanpur, India. His research interests are Power systems, Expert Systems applications, HVDC transmission, Fuzzy logic and Neural Networks application and KARMAA (Knowledge Acquisition, Retention, Management, Assimilation & Application). 12