Coherent updating of non-additive measures

International Journal of Approximate Reasoning, 2016

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of n-coherent and n-convex conditional previsions, at the varying of n. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) 2-convex or, if positive homogeneity and conjugacy is needed, 2-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalized Bayes Rule and always have a 2-convex or, respectively, 2-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, n-convex and n-coherent previsions with n ≥ 3 either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by 2-convexity, we discuss generalizations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered 2convex. In the final part, we determine the rationality requirements of 2-convexity and 2-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.

International Journal of Approximate Reasoning, 2008

This paper presents a summary of Peter Walley's theory of coherent lower previsions. We introduce three representations of coherent assessments: coherent lower and upper previsions, closed and convex sets of linear previsions, and sets of desirable gambles. We show also how the notion of coherence can be used to update our beliefs with new information, and a number of possibilities to model the notion of independence with coherent lower previsions. Next, we comment on the connection with other approaches in the literature: de Finetti's and Williams' earlier work, Kuznetsov's and Weischelberger's work on interval-valued probabilities, Dempster-Shafer theory of evidence and Shafer and Vovk's game-theoretic approach. Finally, we present a brief survey of some applications and summarize the main strengths and challenges of the theory.

Mathematics

In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [0,∞[n is presented. Lastly, some concluding remarks are added.

2005

We study n-monotone lower previsions, which constitute a generalisation of n-monotone lower probabilities. We investigate their relation with the concepts of coherence and natural extension in the behavioural theory of imprecise probabilities, and improve along the way upon a number of results from the literature.

Fuzzy Sets and Systems, 2017

Several consistency notions are available for a lower prevision P assessed on a set D of gambles (bounded random variables), ranging from the well known coherence to convexity and to the recently introduced 2-coherence and 2-convexity. In all these instances, a procedure with remarkable features, called (coherent, convex, 2-coherent or 2-convex) natural extension, is available to extend P , preserving its consistency properties, to an arbitrary superset of gambles. We analyse the 2-coherent and 2-convex natural extensions, E 2 and E 2c respectively, showing that they may coincide with the other extensions in certain, special but rather common, cases of 'full' conditional lower prevision or probability assessments. This does generally not happen if P is a(n unconditional) lower probability on the powerset of a given partition and is extended to the gambles defined on the same partition. In this framework we determine alternative formulae for E 2 and E 2c. We also show that E 2c may be nearly vacuous in some sense, while the Choquet integral extension is 2-coherent if P is, and bounds from above the 2-coherent natural extension. Relationships between the finiteness of the various natural extensions and conditions of avoiding sure loss or weaker are also pointed out.

International Journal of General Systems, 2018

In this paper, it is proven that the natural extensions of a submodular coherent upper conditional probability, defined a class S properly contained in the power set of , coincide on the class of all bounded and upper S-measurable random variables. Moreover, it is proven that a coherent upper conditional prevision can be represented as the Choquet integral, the pan-integral and the concave integral with respect to its associated Hausdorff outer measure h s and, denoted by S the σ -field of all h s -measurable sets, all these integral representations agree with the Lebesgue integral on the class of all S-measurable random variables.

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of n-coherent and n-convex conditional previsions, at the varying of n. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) 2-convex or, if positive homogeneity and conjugacy is needed, 2-coherent lower previsions. Basic properties of these previsions are studied. In particular, centered 2-convex previsions satisfy the Generalized Bayes Rule and always have a 2-convex natural extension. We discuss then the rationality requirements of 2-convexity and 2-coherence from a desirability perspective. Among the uncertainty concepts that can be modelled by 2-convexity, we mention generalizations of capacities and niveloids to a conditional framework.

Journal of Mathematical Analysis and Applications, 2014

We investigate under which conditions a transformation of an imprecise probability model of a certain type (coherent lower previsions, n-monotone capacities, minitive measures) produces a model of the same type. We give a number of necessary and sufficient conditions, and study in detail a particular class of such transformations, called filter maps. These maps include as particular models multi-valued mappings as well as other models of interest within imprecise probability theory, and can be linked to filters of sets and {0, 1}-valued lower probabilities.

Proc. ISIPTA

In this paper we consider some bounds for lower previsions that are either coherent or centered convex. As for coherent conditional previsions, we adopt a structure-free version of Williams’ coherence, which we compare with Williams’ original version and with other coherence concepts. We then focus on bounds concerning the classical product and Bayes’ rules. After discussing some implications of product rule bounds, we generalise a well-known lower bound, which is a (weak) version for coherent lower probabilities of Bayes’ theorem, to the case of (centered) convex previsions. We obtain a family of bounds and show that one of them is undominated in all cases.

International Journal of Approximate Reasoning, 2007

We generalise Walley's Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the so-called forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions.