The use of Markov operators to constructing generalised probabilities

A new approach to constructing generalised probabilities is proposed. It is based on the models using lower and upper previsions, or equivalently, convex sets of probability measures. Our approach uses sets of Markov operators in the role of rules preserving desirability of gambles. The main motivation being the operators of conditional expectations which are usually assumed to reduce riskiness of gambles. Imprecise probability models are then obtained in the ways to be consistent with those desirability preserving rules. The consistency criteria are based on the existing interpretations of models using imprecise probabilities. The classical models based on lower and upper previsions are shown to be a special class of the generalised models. Further, we generalise some standard extension procedures, including the marginal extension and independent products, which can be defined independently of the existing procedures known for standard models.