Abstract
•Measure-based model of uncertain information as generalization of probability.•Consider uncertain joint variables using a measure on the joint space.•Choquet integral of function of joint variable under measure type uncertainty.•Determine mean, variance, covariance and correlation for measure type uncertainty.
We introduce the idea of a measure-based representation of uncertain information as a generalization of probabilistic uncertainty. We look at the situation of uncertain joint variables where our knowledge of the uncertain joint variable is captured by a measure µ on the joint space. We introduce the formulation of the Choquet integral of a function of a joint variable with respect to the measure modeling the underlying uncertainty. This allows us to calculate a mean like value of the joint function with respect to the uncertainty captured by the measure µ. By appropriately selecting the function we are able to extend the formulation of some fundamental concepts used in probabilistic uncertainty to the more general case of measure-based uncertainty. Among the concepts investigated are the means and variances of the individual variables, the covariance and correlation between the joint variables. We also look at the mean and variance of the sum of the joint variables.