Convex Imprecise Previsions for Risk Measurement

In this paper we introduce convex imprecise previsions as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of weakly convex imprecise previsions is also studied and its fundamental properties are demonstrated. The notions of weak convexity and convexity are then applied to risk measurement, leading to a more general definition of convex risk measure than the one already known in risk measurement literature.imprecise previsions, risk measures, weakly convex imprecise previsions, convex imprecise previsions

Convex Imprecise Previsions: Basic Issues and Applications

In this paper we study two classes of imprecise previsions, which we termed convex and centered convex previsions, in the framework of Walley's theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural estension, which is useful in correcting a large class of inconsistent imprecise probability assessments. This class is characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. Finally, we show how these concepts can be applied in (financial) risk measurement.Comment: Proceedings of ISIPTA'0

The Goodman-Nguyen Relation within Imprecise Probability Theory

The Goodman-Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman-Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.Comment: Published version: http://www.sciencedirect.com/science/article/pii/S0888613X1400101

2-coherent and 2-convex Conditional Lower Previsions

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 $n\geq 3$ 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 $2$-convex. 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.Comment: This is the authors' version of a work that was accepted for publication in the International Journal of Approximate Reasoning, vol. 77, October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003, http://www.sciencedirect.com/science/article/pii/S0888613X1630079

Believing Probabilistic Contents: On the Expressive Power and Coherence of Sets of Sets of Probabilities

Moss (2018) argues that rational agents are best thought of not as having degrees of belief in various propositions but as having beliefs in probabilistic contents, or probabilistic beliefs. Probabilistic contents are sets of probability functions. Probabilistic belief states, in turn, are modeled by sets of probabilistic contents, or sets of sets of probability functions. We argue that this Mossean framework is of considerable interest quite independently of its role in Moss’ account of probabilistic knowledge or her semantics for epistemic modals and probability operators. It is an extremely general model of uncertainty. Indeed, it is at least as general and expressively powerful as every other current imprecise probability framework, including lower probabilities, lower previsions, sets of probabilities, sets of desirable gambles, and choice functions. In addition, we partially answer an important question that Moss leaves open, viz., why should rational agents have consistent probabilistic beliefs? We show that an important subclass of Mossean believers avoid Dutch bookability iff they have consistent probabilistic beliefs

Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains

The effect of perturbations of parameters for uniquely convergent imprecise Markov chains is studied. We provide the maximal distance between the distributions of original and perturbed chain and maximal degree of imprecision, given the imprecision of the initial distribution. The bounds on the errors and degrees of imprecision are found for the distributions at finite time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure

Lower and upper covariance

We give a definition for lower and upper covariance in Walley's theory of imprecise probabilities (or coherent lower previsions) that is direct, i.e., does not refer to credal sets. It generalizes Walley's definition for lower and upper variance. Just like Walley's definition of lower and upper variance, our definition for lower and upper covariance is compatible with the credal set approach; i.e., we also provide a covariance envelope theorem. Our approach mirrors the one taken by Walley: we first reformulate the calculation of a covariance as an optimization problem and then generalize this optimization problem to lower and upper previsions. We also briefly discuss the still unclear meaning of lower and upper (co)variances and mention some ideas about generalizations to other central moments

Accept & Reject Statement-Based Uncertainty Models

We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss---as a bridge to existing frameworks---a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.Comment: 35 pages, 17 figure

Jensen's and Cantelli's Inequalities with Imprecise Previsions

We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable $X$, considering either convex/concave functions of $X$ (Jensen's inequalities) or one-sided bounds such as $(X\geq c)$ or $(X\leq c)$ (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably $2$-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds.Comment: Published in Fuzzy Sets and Systems - https://dx.doi.org/10.1016/j.fss.2022.06.02