Inference and risk measurement with the pari-mutuel model

We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance, compare the PMM with a related model, the Total Variation Model, and generalize the natural extension of the PMM introduced by P. Walley and other pertained formulae. The results are subsequently given a risk measurement interpretation: in particular it is shown that a known risk measure, Tail Value at Risk (TVaR), is derived from the PMM, and a coherent risk measure more general than TVaR from its imprecise version. We analyze further the conditions for coherence of a related risk measure, Conditional Tail Expectation. Conditioning with the PMM is investigated too, computing its natural extension, characterising its dilation and studying the weaker concept of imprecision increase.