By Prof. Dr. Jürg Kohlas, Dr. Paul-André Monney (auth.)
The topic of the ebook is an method of the modeling of and the reasoning lower than uncertainty. It develops the Dempster-Shafer idea as a idea of the reliability of reasoning with doubtful arguments. a specific curiosity of this procedure is that it yields a brand new synthesis and integration of common sense and chance concept. The reader will take advantage of a brand new view at uncertainty modeling which extends classical likelihood theory.
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Additional info for A Mathematical Theory of Hints: An Approach to the Dempster-Shafer Theory of Evidence
Sample text
We may also ask to what extent the hypothesis H appears as possible or plausible in view of the hint. 3) f. 0}. 4) We can also forumulate the following theorem. 1. If H, H' and HI/ are subsets of 6, then (1) (2) (3) (4) u(0) = v(0) = 0, u(6) = v(6) = il v(H) = u(HCl, u(H) = v(H c )" u(H) ~ v(H) If H' ~ HI/ then u(H') ~ u(H") and v(H') ~ v(H"). The proof of this theorem, like that of alI theorems in this chapter, can be found in the mathematical appendix at the end of the chapter. This theorem shows that the support and plausibility of the empty hypothesis is zero, whereas these are equal to one for the hypothesis 6.
This evidence can now be combined with the a priori probability O' of a defect. J. Bernoulli. In order to make our process of reasoning clear, let's introduce an argument a which implies d, if a is true, and assume that a has the probability 0'. 6 Imprecise Measurements 27 probability, both a and a failure mode Wj, such that x E T(wj) must be true. The probability of such a proof is k a pl( x l, where k accounts once more for the possibility of contradictory arguments. On the other hand, in order to prove 1, -'a must hold and the error of the functioning device must be x.
R- r. 1. Suppose that the question considered is what is the result of flipping a coin. Then of course 8 = {heads, tails}. Suppose there are several people who tel! us the result; the problem is only that these people may be liars who tel! exactly the contrary of the truth. So, somebody asserting that the coin turned "heads" defines a hint with two possible interpretations Wl and W2. If he tells the truth (interpretation wJ), then r(Wl) = {heads} and if he is a liar (interpretation W2), then r( W2) = {tai ls }.