The properties of negation of a probability distribution recently defined by Yager are studied. Furthermore, the negation of joint and marginal probability distributions in the bivariate case has been defined and their properties are studied. Finally, we have defined a new entropy function for determination of uncertainty associated with the negation of a probability distribution and the events associated with it.
The negation of a probability distribution developed by Yager and Liu deals with the uncertainty when the probability of a particular event is equally distributed among the other alternatives. In this case, intuitively the uncertainty associated with that particular event should be equally distributed among the uncertainty associated with the other alternatives. However, for verifying this, we need to develop uncertainty measures that will measure the uncertainty associated with the negation of a probability distribution. In the present study, we have developed measures analogous to the well known Shannon entropy function and the Kullback Leibler’s divergence for determining the uncertainty related with the negation and studied its properties. Finally, we have derived the Baye’s thoerem associated with the negation of probabilities of a sequence of events.
Negation in general represents the contradiction and denial of anything. In classical logic, negation is identified as the logical connective that takes truthfulness to falsehood. However, there can be situations in which we have to deal with the possibility of any truth value besides true or false. The probability theory has been effective in modelling such situations. Therefore exploring the concept of probability functions from the negation point of view should be helpful in investigating new horizons related with a random phenomena. The negation of a probability distribution results from a transformation which reallocates the probability of each event equally among the other alternatives in a finite sample space. Proposed by Yager, this transformation possesses the maximum entropy allocation. Since any type of mixing increases the disorder(uncertainty) in a system, intuitively the uncertainty associated with the obtained negation must be greater than the uncertainty associated with the original probability distribution. A closer look at Yager's negation reveals that negation of probabilities are convex combinations of the probabilities in the original distribution and we have utilized this for estimating the unpredictability associated with the negation and the implications have been discussed in detail. Also, we have shown that the concept of negation can be helpful for
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