Constructing belief functions from sample data using multinomial confidence regions

The Transferable Belief Model is a subjectivist model of uncertainty in which an agent’s beliefs at a given time are modeled using the formalism of belief functions. Belief functions that enter the model are usually either elicited from experts, or must be constructed from observation data. There are, however, few simple and operational methods available for building belief functions from data. Such a method is proposed in this paper. More precisely, we tackle the problem of quantifying beliefs held by an agent about the realization of a discrete random variable X with unknown probability distribution PX , having observed a realization of an independent, identically distributed random sample with the same distribution. The solution is obtained using simultaneous confidence intervals for multinomial proportions, several of which have been proposed in the statistical literature. The proposed solution verifies two “reasonable” properties with respect to PX : it is less committed than PX with some user-defined probability, and it converges towards PX in probability as the size of the sample tends to infinity. A general formulation is given, and a useful approximation with a simple analytical expression is presented, in the important special case where the domain of X is ordered.