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A somewhat different model is used by the Dempster-Shafer representation. The model has two features: first, it uses a combination calculus of multiplication of each singular observation's confidence with the probability of the phenomenon within the realm of all possible phenomena, and sums up all related results of multiplications. This orthogonal calculation method permits the consideration of all possible combinations of phenomena and causes, respectively (Figure 5).

The method is composed really of two parts: Dempster's rule of combination of evidences and Shafer's application of it for a calculus and philosophy of evidence distribution among certain events and groups of events, events where we have an estimate and those where the chances are completely hidden.

Dempster's rule suggests an orthogonal sum of the estimates in the following way:

A = (A1 ... An) a set of events (symptoms, etc.)

B = (B1 ... Bn) the other set of the same situation

m1(Ai), m2(Bj) are the related estimates; each set of those should be complete , then for the combination (intersection)

The denominator serves for discarding the empty parts of the intersections and for normalization to the measure 1.

The frame of discernment is the totality of events and their combinations within a certain situation (case). The strength of this concept is that it manages combined evidences as, e.g., symptoms that can conclude to more than one disease.

The concept of the frame of discernment permits one to define estimates to all known cases and then combinations and leave the remainder to the whole territory of the unknowns. By combining the different sets of these estimates (different investigations, different witnesses, etc.), the uncertainty of the unknown region (remainder of the frame of discernment) decreases gradually. This is the effect of smearing the evidences through the possible cases.


FIGURE 5 Dempster-Shafer uncertainty representation: orthogonal combinations of information sources and hypotheses on causes.

One certain case of evidence (one symptom, one statement of a witness) is a basic probability assignment (the values in the Dempster method), the belief function is the sum of those for a subset of the frame (Bel A). The plausibility function is a complement:

P1(A) = 1 - Bel(A)

The other feature is the permission of hypothetical unknown phenomena, and a narrowing procedure of this unknown realm, similar to our investigation procedures in the case of several possible hypotheses and unknown causes or effects. The model yields results for these unknown situations, different from the Bayesian, which distributes the probabilities in an equal proportion if there is no definite probability estimation for those. Several recommendations exist in the literature for establishing bridges between the two models. Another weakness of the model is its deformed behavior in case of marginal probability estimations. The strength of the model excels in cases of multiple combined observations and situations, and in case of human judgement instead of statistical data. Typical application areas are those referred to in the beginning of this section: inference in a complex medical diagnosis or criminal investigation.

The other favorite model is the fuzzy representation of Zadeh. The membership estimation of an event, situation in a set of similar cases reflects the pattern metaphor, and simultaneously, the verbal qualification practice of human judgement. The distribution of a membership-judgement class follows also the human practice, we think more in fuzzy way impressions of distribution regions than in the mathematically correct Gaussian-like distribution patterns (Figure 6).

The fuzzy membership value is consciously context-dependent, and has a semantics. The property of attributing this semantics opens the possibility of a direct use for a linguistic interface and for the development of a fuzzy language. The distribution concept of the probability theory has a simplified version by trapezoid- or triangular-like distributions, where the lower and the upper limits of the enclosed area are the possibility (resp. necessity) limits of the set definition. Combination is somehow relative to the same theoretic solutions and this is not be hance: the problem is for both to get a reliable estimate of uncertain events' coincidence.

In the fuzzy notation [mu]F(F) is the membership function of a set F, i.e., [mu]iF expresses how the element i is guessed to be belonging to the set F. as in any similar uncertainty methods. The membership guess can be expressed by word, e.g., high, low, very cold, hot, etc. Both possibilistic and fuzzy logic are built upon this concept.


FIGURE 6 Fuzzy representation.

Possibilistic logic is concerned with uncertain reasoning where the database is a fuzzy description of a given world.

The fuzzy combination rules:

for possibility estimation [pi](u) are defined, the degree of possibility that the datum concerned (the height of Mary or the temperature next day at noon) is exactly equal to u by this definition; two further values can be conceptualized, an upper bound named possiblity ([Pi]) and a lower bound, the necessity (N):

This results in a consciously more subjective classification than any other method claims. The combination-propagation model follows the minimax concepts of game theory, delimiting the outcomes into the possible and necessary extremes. These limits assign rather wide ranges; in practical cases a further fuzzy estimation of the resulting situation is more substantial. The fuzzy representation is useful in all cases where qualitative judgements are common, and not too much more information is available. Similarly, fuzzy representation works well in all cases where the requirement and possibility for more sophisticated methods is not necessary. Fuzzy control is a typical field. Zadeh emphasizes that the fuzzy method has no claim to replace any other, but in many cases it works well and can be applied in combination with others, too.

An important further development of fuzzy-qualitative representation, though not in an acknowledged relation to the fuzzy view, is qualitative reasoning, used especially for describing physical processes (Figure 7).


FIGURE 7 Naive physics -- qualitative reasoning.


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