A note on negation of a probability distribution

Evaluating the negation of an uncertain event is an open issue. Yager (IEEE Trans Fuzzy Syst 23:1899–1902, 2004) suggested a transformation for evaluating the negation of a probability distribution. He used the idea that any event whose outcome is not certain can be negated by supporting the occurrence of other events with no bias or prejudice for any particular outcome. Various authors have tried to generalize the negation transformation proposed by Yager (IEEE Trans Fuzzy Syst 23:1899–1902, 2004). However, we need to focus on developing the basic structure of negation so that the behaviour of the process modelled by negation transformation can be understood in detail. Yager’s negation is based on distribution of maximum entropy. If a probability distribution is uncertain(a state other than maximum entropy), the more the iterations of negation, the more uncertain this probability event becomes, eventually converging to a homogeneous state, i.e. maximum entropy. In other words, it is the realization of the process. What is noted that during each negation, Yager’s method ensures that the negation is intuitive; the next negation weakens the probability of the event occurring in the previous step. Since negation involves reallocation of probabilities at each step in such a way that the reallocation at each step can be determined from the reallocation at the previous step, it is clear that Yager’s negation has various attributes similar to that of a Markov chain. In the present work, we have shown that Yager’s definition of negation can be modelled as a Markov chain which is irreducible, aperiodic with no absorbing states. Two examples have been discussed to strengthen and support the analytical results. Also, we have defined an information generating function (IGF) whose derivative evaluated at specific points gives the moments of the self-information of negation of a probability distribution. The properties of the generating function along with its relationship with the information generating function proposed by S. Golomb (IEEE Trans Inf Theory 12:75–77, 1966) have been explored. A closer look at the properties of IGF confirms the existence of Markovian structure of Yager’s negation.

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The authors did not receive support from any organization for the submitted work. The authors have no relevant financial or non-financial interests to disclose.

Authors and Affiliations

1. Department of Mathematics, Jaypee Institute of Information Technology, Noida, 201304, India

   Manpreet Kaur & Amit Srivastava

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All authors whose names appear on the submission made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data; or the creation of new software used in the work; drafted the work or revised it critically for important intellectual content; approved the version to be published; and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Corresponding author

Correspondence to Amit Srivastava.

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