Normal-Wishart Distribution based on Fuzzy Interval Data －1st Report: Theoretical analysis and numerical experiments for models with unknown average vector and variance-covariance matrix－

AbstractIn this research, we derive probability functions for normal-Wishart distributions based on fuzzy interval data obtained from normal probability space.A multivariate normal distribution is assumed, and its mean vector and variance-covariance matrix are unknown.From the distribution, a random variable vector is given as fuzzy interval data with ambiguity.Here, we proceed with the formulation based on the concept of probability of fuzzy events proposed by Zadeh.Fuzzy interval data are characterized using membership functions.Three types of membership functions are formulated.However, the drawback is that the integrals of the products of these and the probability density functions of the normal-Wishart distributions are analytically very complicated.In this situation, the method proposed in this research, which uses the central value of the membership function as a representative value, allows us to calculate the posterior probability without much change from the case of ordinary accurate random variables.This allows for easy observation of fuzzy interval data.However, since representative values alone can cause bias, a correction amount is derived to eliminate this.Numerical examples are presented to illustrate the theoretically derived posterior probability functions.