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## Publications

Publications (73)

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 $X$ (Jense...

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...

Dilation is a puzzling phenomenon within Imprecise Probability theory: when it obtains, our uncertainty evaluation on event $A$ is vaguer after conditioning $A$ on $B$, whatever is event $B$ in a given partition $\mathcal{B}$. In this paper we investigate dilation with coherent Nearly-Linear (NL) models. These are a family of neighbourhood models,...

Dilation is a puzzling phenomenon within Imprecise Probability theory: when it obtains, our uncertainty evaluation on event A is vaguer after conditioning A on B, whatever is event B in a given partition B. In this paper we investigate dilation with coherent Nearly-Linear (NL) models. These are a family of neighbourhood models, obtaining lower/uppe...

Coherent lower probabilities are one of the most general tools within Imprecise Probability Theory, and can be used to model the available information about an unknown or partially known precise probability. In spite of their generality, coherent lower probabilities are sometimes difficult to deal with. For this reason, in previous papers we studie...

The process of outer approximating a coherent lower probability by a more tractable model with additional properties, such as 2- or completely monotone capacities, may not have a unique solution. In this paper, we investigate whether a number of approaches may help in eliciting a unique outer approximation: minimising a number of distances with res...

In previous work [1] we introduced Nearly-Linear (NL) models, a class of neighbourhood models obtaining upper/lower probabilities by means of a linear affine transformation (with barriers) of a given probability. NL models are partitioned into more subfamilies, some of which are coherent. One, that of the Vertical Barrier Models (VBM), includes kno...

Several simplified uncertainty models are derived from a given probability P0 of which they are a perturbation. Among these, we introduced in previous work Nearly-Linear (NL) models. They perform a linear affine transformation of P0 with barriers, obtaining a couple of conjugate lower/upper probabilities, and generalise several well known neighbour...

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P_0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact...

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact,...

Nearly-Linear Models are a family of neighbourhood models, obtaining lower/upper probabilities from a given probability by a linear affine transformation with barriers. They include a number of known models as special cases, among them the Pari-Mutuel Model, the ε-contamination model, the Total Variation Model and the vacuous lower/upper probabilit...

From an epistemic point of view, coherent lower probabilities allow us to model the imprecise information about a partially unknown probability. However, there are some issues that hinder their use in practice. Since belief functions are easier to deal with, we propose to approximate the coherent lower probability by a belief function that is at th...

We introduce two models for imprecise probabilities which generalise the Pari-Mutuel Model while retaining its simple structure. Their consistency properties are investigated, as well as their capability of formalising an assessor’s different attitudes. It turns out that one model is always coherent, while the other is (occasionally coherent but) g...

We investigate the problem of outer approximating a coherent lower probability with a more tractable model. In particular, in this work we focus on the outer approximations made by belief functions. We show that they can be obtained by solving a linear programming problem. In addition, we consider the subfamily of necessity measures, and show that...

We investigate the problem of approximating a coherent lower probability on a finite space by a 2-monotone capacity that is at the same time as close as possible while not including additional information. We show that this can be tackled by means of a linear programming problem, and investigate the features of the set of undominated solutions. Whi...

We investigate the problem of approximating a coherent lower probability on a finite space by a 2-monotone capacity that is at the same time as close as possible while not including additional information. We show that this can be tackled by means of a linear programming problem, and investigate the features of the set of undominated solutions. Whi...

Several consistency notions for lower previsions (coherence, convexity, others) require that the suprema of certain gambles, having the meaning of gains, are non-negative. The limit situation that a gain supremum is zero is termed Weak Dutch Book (WDB). In the literature, the special case of WDBs with precise probabilities has mostly been analysed,...

Combining computational models of argumentation with probability theory has recently gained increasing attention, in particular with respect to abstract argumentation frameworks. Approaches following this idea can be categorised into the constellations and the epistemic approach. While the former considers probability functions on the subgraphs of...

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 dedicate...

Combining computational models of argumentation with probability theory has recently gained increasing attention, in particular with respect to abstract argumentation frameworks. Approaches following this idea can be cate-gorised into the constellations and the epistemic approach. While the former considers probability functions on the subgraphs of...

Several consistency notions are available for a lower prevision assessed on a set 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) natura...

Several consistency notions for lower previsions (coherence, convexity, others) require that the suprema of certain gambles, having the meaning of gains, are non-negative. The limit situation that a gain supremum is zero is termed Weak Dutch Book (WDB). In the literature, the special case of WDBs with precise probabilities has mostly been analysed,...

The recently introduced
weak consistency notions
of 2-coherence and 2-convexity are endowed with a concept of 2-coherent, respectively, 2-convex natural extension, whose properties parallel those of the natural extension for coherent lower previsions. We show that some of these extensions coincide in various common instances, thus producing the sam...

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 i...

Constrained coherence is compared to coherence and its role in the behavioural interpretation of coherence is discussed. The equivalence of these two notions is proven for coherent conditional previsions, showing that the same course of reasoning applies to several similar concepts developed in the realm of imprecise probability theory .

A p-box is a simple generalization of a distribution function, useful to study a random number in the presence of imprecision. We propose an extension of p-boxes to cover imprecise evaluations of pairs of random numbers and term them bivariate p-boxes. We analyze their rather weak consistency properties, since they are at best (but generally not) e...

Sklar's theorem is an important tool that connects bidimensional distribution
functions with their marginals by means of a copula. When there is imprecision
about the marginals, we can model the available information by means of
p-boxes, that are pairs of ordered distribution functions. Similarly, we can
consider a set of copulas instead of a singl...

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 i...

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 al...

The theory of imprecise probabilities offers potentially many applications in the realms of finance and economics. The focus in this chapter is on a few applications. It takes a closer look at the betting scheme in the definition of coherence, partly to discuss how the betting scheme could be applied in real-world bets. The chapter illustrates the...

Epistemic probabilities in argumentation frameworks are meant to represent subjective degrees of belief in the acceptance of arguments. As such, they are subject to some rationality conditions, taking into account the attack relation between arguments. This paper provides an advancement with respect to the previous literature on this matter by cast...

Sklar's theorem is an important tool that connects bidimensional distribution functions to their marginals by means of a copula. When there is imprecision about the marginal models, we can model the available information by means of p-boxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a s...

The Goodman-Nguyen relation generalises the implication (inclusion) relation to conditional events. As such, it induces inequality constraints relevant in extension problems with precise probabilities. We extend this framework to imprecise probability judgements, highlighting the role of this relation in determining the natural extension of lower/u...

We review several of de Finetti’s fundamental contributions where these have played and continue to play an important role in the development of imprecise probability research. Also, we discuss de Finetti’s few, but mostly critical remarks about the prospects for a theory of imprecise probabilities, given the limited development of imprecise probab...

We review several of de Finetti's fundamental contributions where these have played and continue to play an important role in the development of imprecise probability research. Also, we discuss de Finetti's few, but mostly critical remarks about the prospects for a theory of imprecise probabilities, given the limited development of imprecise probab...

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...

We explore some little investigated aspects of the well known betting scheme defining coherent lower or upper previsions in
terms of admissible gains. A limiting situation (lose-or-draw) where the supremum of some gain is zero is discussed, deriving
a gambler’s gain evaluations and comparing the differences between the imprecise and precise previsi...

In this paper we discuss the consistency concept of Williams coherence for
imprecise conditional previsions, presenting a variant of this notion, which we
call W-coherence. It is shown that W-coherence ensures important consistency
properties and is quite general and well-grounded. This is done comparing it
with alternative or anyway similar known...

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 wit...

In this paper we consider some bounds for lower previsions that are either coherent or, more generally, centered convex. We focus on bounds concerning the classical product and Bayes' rules, discussing first weak product rules and some of their implications for coherent lower previsions. We then generalise a well-known lower bound, which is a (weak...

We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assess-ing an upper probability from a precise one. We dis-cuss a naive extension of the PMM considered in in-surance and generalize the natural extension of the PMM introduced by P. Walley and other related for-mulae. The results are subsequently give...

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 measur...

In this paper we consider some bounds for lower pre-visions 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 Ba...

These notes comment on Williams’ fundamental essay Notes on Conditional Previsions, written as a research report in 1975 and published in the present issue. Basic aspects of that work are discussed, including historical background and relevance to the foundations of probability; examples are supplied to help understanding.

Because of their simplicity, risk measures are often employed in financial risk evaluations and related decisions. In fact, the risk measure (X) of a random variable X is a real number customarily determining the amount of money needed to face the potential losses X might cause. At a sort of second-order level, the adequacy of (X) may be investigat...

The problem of computing the maximum-entropy precise probabil- ity consistent with a 2-monotone ca- pacity is solved by an algorithm de- vised by Jaffray. After reviewing its properties, with particular refer- ence to its applicability scope, we introduce a modified version of the algorithm, showing that it works correctly on a strictly larger fami...

This paper addresses the problem of exchanging uncertainty assessments in multi-agent systems. Since it is assumed that each agent might completely ignore the internal representation of its partners, a common interchange format is needed. We analyze the case of an interchange format defined by means of imprecise probabilities, pointing out the reas...

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 estension, which is useful in correcting a large class
of inconsistent imprec...

This paper focuses on establishing envelope theorems for convex conditional lower previsions, a recently in- vestigated class of imprecise previsions larger than co- herent imprecise conditional previsions. It is in partic- ular discussed how the various theorems can be em- ployed in assessing convex previsions. We also con- sider the problem of di...

We solve two fundamental problems of probabilistic reasoning: given finitely many conditional probability assessments, how to determine whether the assessments are mutually consistent, and how to determine what they imply about the conditional probabilities of other events? These problems were posed in 1854 by George Boole, who gave a partial solut...

In this paper we analyze, mainly in a finitary setting, the consistency properties of fuzzy possibilities, interpreting them as instances of upper previsions and applying the basic notions of avoiding sure loss and coherence from the theory of imprecise probabilities. It ensues that fuzzy possibilities always avoid sure loss, but satisfy the strong...

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 ar...

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 propert...

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...

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 asses...

Several known procedures transforming an imprecise probability into a precise one focus on special classes of imprecise probabilities,
like belief functions and 2–monotone capacities, while not addressing the more general case of coherent imprecise probabilities,
as defined by Walley. In this paper we first analyze some of these transformations, ex...

Cited By (since 1996): 3, Export Date: 31 January 2012, Source: Scopus, CODEN: IJUSF, doi: 10.1142/S0218488503002156, Language of Original Document: English, Correspondence Address: Pelessoni, R.; Dipto. Matemat. Applicata B. Finetti, University of Trieste, Piazzale Europa 1 Trieste, I-34127, Italy; email: renato.pelessoni@econ.units.it, References...

The interpretations of belief functins and their relationships with other uncertainty theories have been widely debated in
the literature. Focusing on the interpretation of belief functions based on non-negative masses, in this paper we provide
a contribution to this topic by addressing two questions concerning the relationships between belief func...

In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (VaR), are studied from the perspective of the theory of coherent imprecise previsions. 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...

The aim of this paper is that of studying a notion of independence for imprecise probabilities which is essentially based on the intuitive meaning of this concept. This is expressed, in the case of two events, by the reciprocal irrelevance of the knowledge of the value of each event for evaluating the other one, and has been termed epistemic indepe...

The aim of this paper is that of studying a notion of independence for imprecise probabilities which is essentially based on the intuitive meaning of this concept. This is expressed, in the case of two events, by the reciprocal irrelevance of the knowledge of the value of each event for evaluating the other one, and has been termed epistemic indepe...

In this paper we introduce an operational procedure which, given an avoiding sure loss (ASL) imprecise probability assessment on an arbitrary finite set of conditional events, determines its 'least-committal' coherent correction, i.e. the coherent imprecise probability assessment which reduces the imprecision of as little as possible, without ever...

A definition of realization for comparative probabilities using coherent conditional probabilities is proposed in a finite setting and named stratified realization. The extent of stratified realization is examined in detail and comparisons are made with realization by means of a strictly positive, not necessarily bounded real measure (orP
1-realiza...

Some features of the connection between comparative and quantitative probabilities are examined in the context of a probabilistic approach to Artificial Intelligence. Coherent conditional probabilities are used: this makes it possible to work with sets of events which are not necessarily ‘structured’, but which arise in a natural way from the real...

In multi-agent systems, there is the need to exchange uncertain information between distinct and independently developed software components. This requires that such components share a common uncertainty interchange format and poses, therefore, a serious and still poorly considered problem, in face of the variety of existing uncertainty theories. I...