Extensions of belief functions and possibility distributions by using the imprecise Dirichlet model

A belief function can be viewed as a generalized probability function and the belief and plausibility measures can be regarded as lower and upper bounds for the probability of an event. However, the classical probabilistic interpretation used for computing belief and plausibility measures may be unreasonable in many real applications when the number of observations or measurements is rather small. In order to overcome this difficulty, Walley's imprecise Dirichlet model is used to extend the belief, plausibility and possibility measures. An interesting relationship between belief measures and sets of multinomial models is established. Combination rules taking into account reliability of sources of data are studied. Various numerical examples illustrate the proposed extension.