Approximate Bayesian Interval Estimation by Fuzszy Interval Data from Normal Population

Abstract

The qualitative data which contain human consciousness and mental states involve human vagueness, but applications of fuzzy systems theory are effective on such data. In this paper, by using Zadeh's probability concept of fuzzy events, we define the fuzzy interval data and introduce the Bayesian posterior distribution by fuzzy data. However, the method with direct usage of the membership functions of fuzzy interval data treating the membership functions precisely is inefficient from a view point of calculation. But our method treating the middle points of membership functions as the representative points can settle such problems. Here, we suppose that the fuzzy interval data are obtained from normal population. When prior distribution of population parameter θ is normal distribution, we can show that the posterior distribution forms the normal distribution approximately by using our proposed method. As a result, even if we obtain fuzzy interval data, we can explain that the approximate Bayesian interval estimation which is not so far different from the usual Bayesian interval estimation of population parameter θ is possible. In real situations, we do not always obtain ideal symmetrical membership functions. Then, we perform the computer simulations under relistic situations which do not satisfy completely the condition of the symmetry of trapezoidal membership function and examine the practicability of our method. As a result, we can show the practicability.