Imprecise Previsions Research Articles

Bhatia-Davis inequalities for lower and upper previsions and covariances

We explore how the classical Bhatia-Davis inequality bounding variances can be extended to uncertainty evaluations of gambles (bounded random numbers) by means of imprecise (lower or upper) previsions with different degrees of consistency. Firstly, a number of extensions are found with 2-coherent imprecise previsions. Subsequently, bounds with coherent lower and upper previsions are investigated, together with applications bounding lower and upper variances as well as p-boxes. Finally, bounds for covariances and for lower and upper covariances are obtained. Like the classical situation, imprecise Bhatia-Davis inequalities require a reduced amount of uncertainty information to be applied. When even less information is available, we show that various versions of Popoviciu's inequality obtain.

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≥c) or (X≤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.

Weak Dutch Books with imprecise previsions

Uncertainty assessments for imprecise previsions based on coherence and related concepts require that the suprema of certain random numbers (interpreted as gains) are non-negative. The extreme situation that a supremum is zero represents what is called a Weak Dutch Book (WDB) in a betting interpretation language. While most of the previous dedicated literature focused on WDBs for de Finetti's coherence with precise probabilities, in this paper we analyse the properties of WDBs with imprecise previsions, notably for conditional (Williams') coherent lower previsions. We show that WDB assessments ensure a certain ‘local precision’ property and imply, in the agent's evaluation, some kind of ‘protection’ against real losses. Further, these properties vary with the consistency notion we adopt, tending to vanish with weaker ones. A generalisation of the classical strict coherence and other alternative approaches to WDBs are also discussed.

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.

GENERALIZING DUTCH RISK MEASURES THROUGH IMPRECISE PREVISIONS

Relationships between risk measures and imprecise probability theory have received relatively limited attention in the literature. This paper contributes to filling this gap as far as Dutch risk measures are concerned. Using imprecise previsions as a starting point, a novel generalized family of Dutch risk measures is introduced, its properties with respect to several alternative consistency notions are analyzed and its advantageous features discussed. Any such measure corresponds to an imprecise prevision correcting a first-approach uncertainty measure, while preserving its consistency properties in several cases, ranging from coherence to a weak generalization of the concept of capacity. Further, it is shown that the proposed family of measures has a practical significance in the application context of insurance pricing, since, unlike the original formulation, it may ensure a risk loading to each risk taker and is even compatible with the practice of double loading in premium policies.

Importance sampling for Bayesian sensitivity analysis

We examine the use of sensitivity analysis with a particular focus on calculating the bounds of imprecise previsions in Bayesian statistics. We explain the use of importance sampling in approximating the range of these imprecise previsions and we develop an approximation function for the imprecise posterior prevision based on generating a finite number of random variables. We develop a convergence theorem that shows that this approximation converges almost surely to the posterior prevision as we generate more and more random variables. We also develop a useful accuracy bound for the approximation for a large finite number of generated random variables. We test the efficiency of this approximation using a simple example involving the imprecise Dirichlet model.

Financial risk measurement with imprecise probabilities

Although financial risk measurement is a largely investigated research area, its relationship with imprecise probabilities has been mostly overlooked. However, risk measures can be viewed as instances of upper (or lower) previsions, thus letting us apply the theory of imprecise previsions to them. After a presentation of some well known risk measures, including Value-at-Risk or VaR, coherent and convex risk measures, we show how their definitions can be generalized and discuss their consistency properties. Thus, for instance, VaR may or may not avoid sure loss, and conditions for this can be derived. This analysis also makes us consider a very large class of imprecise previsions, which we termed convex previsions, generalizing convex risk measures. Shortfall-based measures and Dutch risk measures are also investigated. Further, conditional risks can be measured by introducing conditional convex previsions. Finally, we analyze the role in risk measurement of some important notions in the theory of imprecise probabilities, like the natural extension or the envelope theorems.

Uncertainty modelling and conditioning with convex imprecise previsions

Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to Walley’s and Williams’ theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural extension, which is useful in correcting a large class of inconsistent imprecise probability assessments, 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. We discuss in particular their usage in (financial) risk measurement. In a final part we introduce convex imprecise previsions in a conditional environment and investigate their basic properties, showing how several of the preceding notions may be extended and the way the generalised Bayes rule applies.

IMPRECISE PREVISIONS FOR RISK MEASUREMENT

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (VaR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called 'avoiding sure loss'. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones.

Variety of judgements admitted in imprecise statistical reasoning

Intending the present paper as a contribution towards precautionary decision-making, the authors give a brief overview and summary of their experience in the modelling of different judgements admitted within the framework of the theory of coherent imprecise previsions and of possible use in reliability analysis. The paper specifically focuses on direct, comparative, structural and unreliable judgements.

Convex imprecise previsions

In this paper centered convex previsions are introduced 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 convex imprecise previsions is also studied and its fundamental properties are demonstrated, introducing in particular a notion of convex natural extension which parallels that of natural extension but has a larger domain of applicability. These concepts appear to have potentially many applications. In this paper they are applied to risk measurement, leading to a general definition of convex risk measure which corresponds, when its domain is a linear space, to the one recently introduced in risk measurement literature.

Processing unreliable judgements with an imprecise hierarchical model

In processing expert judgements, it is common to assign experts’ opinions different degrees of plausibility or probability. This results in the necessity of constructing some kind of hierarchical model. Hierarchical models are widely used in Bayesian inference. Several models have been proposed by the adherents of imprecise statistical reasoning. This paper describes a new hierarchical uncertainty model of a more general form: first- and second-order uncertainty models are both imprecise. We demonstrate that depending on what kind of evidence the modeller has at hand and the reliability quantifying the quality of expert judgement one comes to different rules of combination. It is shown that the rules of combination known in imprecise statistical reasoning can be inferred from the approach proposed in the paper. We describe how the model can be used to generate first-order coherent interval-valued previsions. The paper is concluded by addressing the issue of computing the probability of an event ‘one imprecise value is greater than another’. This case is prominent in light of decision making based on imprecise previsions.

Imprecise predictive selection based on low structure assumptions

Abstract Based on observations of real-valued random quantities from k⩾2 independent sources, new results are presented on bounds for predictive probabilities for further unknown random quantities from each source. Past and future observations per source are related via the assumption A(n) (Hill, J. Amer. Statist. Assoc. 63 (1968) 677), which is closely linked to finite exchangeability. Attention is particularly focussed on the question: which source will provide the largest next observation, when taking one more observation from each source? Our approach enables a nonparametric predictive comparison between the sources, which might naturally be linked to choices between sources, leading us to suggest our method as an alternative formulation to classical selection approaches and to introduce the term ‘imprecise predictive selection’. We present imprecise predictive selection of a single best source and of a subset of m (m⩽k−1) sources. We consider two cases for selection of a subset, namely a subset that contains the m best sources, and a subset that is just required to include the single best source. We also briefly present a related approach, using imprecise previsions with an interpretation as bounds for expected values for future observations.

Comparing two populations based on low stochastic structure assumptions

This paper presents a method to compare two populations without assuming conditional independence for the random quantities in each population. The approach is based on finite exchangeability assumptions per population and is predictive by nature. We present comparison based on imprecise previsions for future observations as well as comparison based on predictive imprecise probabilities. The method is inductive, and can be used when there is not any other relevant knowledge about the random quantities available than the information provided by the data, or if one explicitly does not want to use such knowledge.

On Indifference Zone Selection with a Preference Threshold

AbstractThis paper discusses a selection criterion that generalizes the well‐known concept of indifference zone selection through a preference threshold. A population is preferred to another population if the difference in the sums of observed values exceeds a given nonnegative threshold value. We present an argument for this selection rule by modelling preference by imprecise previsions. We aim at guidelines to design a selection experiment, which is characterized by two numbers: the number of necessary observations per population, and the preference threshold. Next to the probability of correct selection we also need a second specification. In this paper we consider a probability of false selection that is strongly related to the minimum probability of correct selection. Based on this model the outcome of an experiment may be ‘no selection’, at least not based on strong preference of a single population. The ideas are presented through a simple selection problem for normal populations with common known variance. Although the theory has a frequentist nature, the derivation and justification of the selection rule through imprecise previsions relies on Bayesian foundations, and via this route we gain more insight into the selection criterion.