Sets of desirable gambles: Conditioning, representation, and precise probabilities

The CONEstrip Algorithm

“…The work of Walley et al [10] made me believe such an efficient algorithm was possible, the representation of Couso and Moral [3] provided useful structure, and a variable-bounding technique spotted in the 'zero norm'-minimization literature [7, (1) to (2)] made everything come together. The CONEstrip algorithm-formulated in terms of linear programs-is rather high-level.…”

Representation theorems for partially exchangeable random variables

“…Given the discussion in Section 2.5, this representation Q n G is the expectation operator with respect to a probability mass function on the count vectors in N n G , so we obtain in this way the usual finite representation theorem for partial exchangeability as a special case. 16 By combining this special case with Proposition 8(iii), we see that, in general, Q n G is the lower envelope of the count representations Q n G of the linear previsions P G(J ) that dominate P G(J ) . Hence, we find that within our imprecise-probabilistic context, we no longer have a single representing probability mass function on N n G , but rather a (convex) set of them.…”

“…In the introduction, we already mentioned a number of general advantages of sets of desirable gambles, some of which they share with lower previsions. However, there is one feature that they do not share: a set of desirable gambles can always be conditioned in a unique way, even if the conditioning event has (lower) probability zero [16,19]. 26 Consequently, our representation theorems for sets of desirable gambles lead to a representation-or 'prior'-that truly represents all relevant information about a sequence of partially exchangeable variables, including all conditional models.…”

“…We say that V n G (Σ G ) is the set of polynomial gambles of degree up to n G . 16 For partially exchangeable binary random variables, a finite version of de Finetti's representation theorem can be found in Refs. [ V n G (Σ G ) and V(Σ G ) are both linear subspaces of G(Σ G ) that include the constant gambles.…”

An Imprecise Probability Approach to Joint Extensions of Stochastic and Interval Orderings

“…The expert initial information is assessed by means of comparative preference statements between gambles and, afterwards, a set of joint feasible linear previsions or, equivalently, a pair of lower and upper previsions on the set of gambles is derived from it. This is the approach followed in the general theory of imprecise probabilities (see [3,4,20]). W2.…”