A Note on Negation of a Probability Distribution

Abstract Evaluating the negation of an uncertain event is an open issue. Yager[14] 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 occurence of other events with no bias or prejudice for any particular outcome. Various authors have tried to generalize the negation transformation proposed by Yager[14] . 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 iterations of negation, the more uncertain this probability event becomes eventually converging to a homogeneous state\textit{ 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 occuring 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, therefore 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 has 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[2] has been explored. A closer look at the properties of IGF confirms the existence of markovian structure of Yager's Negation.Mathematics Subject Classification (2010) 90C70 · 90B06 · 90B50 · 90C05