Maximum likelihood estimation from fuzzy data using the EM algorithm

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observeddata likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the Fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.