**Papers in proceedings of international conferences**

E. Miranda, G. de Cooman. *Independent
products of numerical possibility measures. *
Proceedings of ISIPTA'01. Ithaca (New York), June 2001.

Abstract: Possibility measures can be given a behavioural interpretation as systems of upper betting rates. As such, they should arguably satisfy certain rationality requirements. Using a version of Walley's notion of epistemic independence suitable for possibility measures, we investigate what these requirements tell us about the construction of independent product possibility measures from given marginals.

E. Miranda, I. Couso, P. Gil. *On the
probabilities dominated by a 2-alternating capacity on a separable metric space.*
Proceedings of AGOP'01. Oviedo (Spain), July 2001.

Abstract: The extreme points of a convex set can be used to characterize its behavior on a wide number of contexts. In this paper, we extend some well-known results about the probabilities dominated by a 2-alternating capacity from the finite to the general case of separable metric spaces. Finally, a closer look to the case of possibility measures is given.

E. Miranda, G. de Cooman, I. Couso. *
Imprecise probabilities induced by multi-valued mappings. *
Proceedings of IPMU'2002. Annecy
(France), July 2002.

Abstract: We discuss how lower previsions induced by multi-valued mappings fit into the framework of the behavioural theory of imprecise probabilities, and show how the notions of coherence and natural extension from that theory can be used to prove and generalise existing results in an elegant and straightforward manner. This provides a clear example for the explanatory and unifying power of these notions.

E. Miranda, I. Couso, P. Gil. *Upper
probabilities and selectors of random sets. *
Proceedings of SMPS'02. Warsaw (Poland), September 2002.

Abstract: We investigate the probabilistic information given by a random set when it represents the imprecise observation of a random variable. We compare the information given by the distributions of the selectors with that provided by the upper and lower probabilities induced by the random set. In particular, we model the knowledge on both the probability of an event and the probability distribution of the original random variable. Some

characterizations and examples are given for the case of a finite final space, and the main difficulties for the infinite case are commented.

E. Miranda, I. Couso, P. Gil. *Study
of the probabilistic information of a random set. *
Proceedings of ISIPTA'03. Lugano (Switzerland), July 2003

Abstract: Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the distribution of this random variable. Specifically, we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the distributions of the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.

Gert de Cooman, Enrique Miranda. A weak law of large numbers for coherent lower previsions. Proceedings of the Tenth International Symposium on Information Processing and Managament of Uncertainty (IPMU'2004). Perugia (Italy), July 2004.

Abstract: We prove a weak law of large numbers for coherent lower previsions. The law is a consequence of the rationality criterion of coherence, and it can be proven under surprisingly weak assumptions. Our treatment also uncovers an interesting connection between the behavioural theory of coherent lower previsions, and Shafer and Vovk's game-theoretic approach to probability theory.

Inés Couso, Enrique Miranda, Gert de Cooman. A possibilistic interpretation of the expectation of a fuzzy random variable. Proceedings of SMPS'2004. Oviedo (Spain), September 2004.

Abstract: We follow a possibilistic interpretation of fuzzy random variables, regarding them as sets of conditional possibility measures. In this vein, we relate them with other models in the context of imprecise probabilities. Our work allows us to deduce a definition of the expectation of a fuzzy random variable and we prove that it coincides with the mean value of the fuzzy expectation proposed by Puri and Ralescu.

Enrique Miranda. Consonant random sets: structure and properties. Proceedings of ECSQARU'05. Barcelona (Spain), July 2005.

In this paper, we investigate consonant random sets from the point of view of lattice theory. We introduce a new definition of consonancy and study its relationship with possibility measures as upper probabilities. This allows us to improve a number of results from the literature. Finally, we study the suitability of consonant random sets as models of the imprecise observation of random variables.

Gert de Cooman, Matthias Troffaes, Enrique Miranda. n-Monotone lower previsions and lower integrals. Proceedings of ISIPTA'05. Pittsburgh (Pennsylvania), July 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. Finally, we indicate how many approaches to integration in the literature fall squarely within the framework of the present study of coherent n-monotone lower previsions. This discussion allows us to characterise which types of integrals can be used to calculate the natural extension of a probability charge.

Enrique Miranda, Gert de Cooman, Erik Quaeghebeur. The moment problem for finitely additive probabilities. Proceedings of IPMU'2006. Paris (France), July 2006.

We study the moment problem for finitely additive probabilities and show that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.

Enrique Miranda, Marco Zaffalon. Coherence graphs. Proceedings of ISIPTA'07. Prague (Czech Republic), July 2007.

In this paper, we study the coherence of the assessments provided by different sources, when these assessments are represented by means of conditional lower previsions. We prove that it is possible to characterise their coherence by means of the properties of an associated graphical representation, called the coherence graph. In doing so, we show that there are some minimal pieces of information whose coherence guarantees the coherence of all the assessments.

Gert de Cooman, Enrique Miranda, Erik Quaeghebeur. Immediate prediction with representation insensitivity. Proceedings of ISIPTA'07. Prague (Czech Republic), July 2007.

We consider immediate predictive inference, where a subject, using a number of observations of a finite number of exchangeable random variables, is asked to coherently model his beliefs about the next observation, in terms of a predictive lower prevision. We study when such predictive lower previsions are representation insensitive, meaning that they are essentially independent of the choice of the (finite) set of possible values for the random variables. Such representation insensitive predictive models have very interesting properties, and among such models, the ones produced by the Imprecise Dirichlet (Multinomial) Model are quite special in a number of ways.

Gert de Cooman, Enrique Miranda, Erik Quaeghebeur. Representing and assessing exchangeable lower previsions. Proceedings of the 56th ISI session. Lisbon (Portugal), August 2007.

This paper deals with belief models, and in particular lower previsions, for both (finite) collections and (infinite) sequences of exchangeable random variables taking a finite number of values. When such collections of sequences are assumed to be exchangeable, this more or less means that their specific order is irrelevant. We show that exchangeable lower previsions can be written as a combination of (i) a coherent prevision expressing that permutations of realisations of such collections or sequences are considered to be equally likely, and (ii) a coherent lower prevision for the 'frequency' of occurrence of the different values the random variables can take. This is the essence of representation in de Finetti's sense: we generalise his results to coherent lower previsions, both for finite collections and infinite sequences. We also solve a more practical problem: how to extend a number of lower prevision assessments to an exchangeable lower prevision that is as conservative as possible.

Enrique Miranda. Coherent updating on finite spaces. Proceedings of IPMU'2008. Málaga (Spain), June 2008.

We compare the different notions of coherence within the behavioural theory of imprecise probabilities when all the spaces are finite. We show that the differences between the notions are due to conditioning on sets of (lower, and in some cases upper) probability zero. Next, we characterise the range of coherent extensions in the finite case, proving that the greatest coherent extensions can always be calculated using the notion of regular extension.

Gert de Cooman, Enrique Miranda. The F. Riesz representation theorem and finite additivity. Proceedings of SMPS'2008. Toulouse (France), September 2008.

A positive and
normalised real linear functional on the set of bounded continuous functions can
be characterised as the integral of a sigma-additive probability measure, by the
F. Riesz Representation Theorem. In this paper, we look at the finitely additive
extensions of such a functional to the set of all bounded random variables, and
prove that they are determined by Riesz' extension to lower semi-continuous
functions. In doing so, we establish links with Daniell's approach to
integration, Walley's theory of coherent lower previsions, and de Finetti's
Representation Theorem for exchangeable random variables.

Enrique Miranda, Matthias Troffaes, Sebastien Destercke. Generalised p-boxes on totally ordered spaces. Proceedings of SMPS'2008. Toulouse (France), September 2008.

Probability boxes are among the most simple and popular models used in imprecise probability theory, and many practical results concerning them exist in the literature. Nevertheless, little attention has been paid to their formal characterisation in the setting of Walley's behavioural theory of imprecise probabilities. This paper tries to remedy this situation by formalising, generalising and extending existing results as well as by giving new ones, within Walley's framework.

Enrique Miranda, Inés Couso, Pedro Gil. Upper probabilities attainable by distributions of measurable selections. Proceedings of ECSQARU'2009. Verona (Italy), July 2009.

A random set can be
regarded as the result of the imprecise observation of a random variable.
Following this interpretation, we study to which extent the upper and lower
probabilities induced by the random set keep all the information about the
values of the probability distribution of the random variable. We link this
problem to the existence of selectors of a multi-valued mapping and with the
inner approximations of the upper probability, and prove that under fairly
general conditions (although not in all cases), the upper and lower
probabilities are an adequate tool for modelling the available information.
Finally, we study the particular case of

consonant random sets and we also derive a relationship between Aumann and
Choquet integrals.

Alessio Benavoli, Marco Zaffalon, Enrique Miranda. Reliable hidden Markov model filtering through coherent lower previsions. Proceedings of FUSION'2009. Seattle (Washington), July 2009.

We extend Hidden Markov Models for continuous variables taking into account imprecision in our knowledge about the probabilistic relationships involved. To achieve that, we consider sets of probabilities, also called coherent lower previsions. In addition to the general formulation, we study in detail a particular case of interest: linear-vacuous mixtures. We also show, in a practical case, that our extension outperforms the Kalman filter when modelling errors are present in the system.

Enrique Miranda, Marco
Zaffalon. Natural
extension as a limit of regular extensions*. *Proceedings of ISIPTA'09.
Durham (England), July 2009.

This paper is devoted to the extension of conditional assessments that satisfy some consistency criteria, such as weak or strong coherence, to further domains. In particular, we characterise the natural extension of a number of conditional lower previsions on finite spaces, by showing that it can be calculated as the limit of a sequence of conditional lower previsions defined by regular extension. Our results are valid for conditional lower previsions with non-linear domains, and allow us to give an equivalent formulation of the notion of coherence in terms of credal sets.

Gert de Cooman, Enrique Miranda and Marco Zaffalon. Independent natural extension. Proceedings of IPMU'2010. Dortmund (Germany), June 2010.

We introduce a general definition for
the independence of a number of finite-valued variables, based on coherent lower
previsions. Our definition has an epistemic flavour: it arises from personal
judgements that a number of variables are irrelevant to one another. We show
that a number of already existing notions, such as strong independence, satisfy
our definition. Moreover, there always is a least-committal independent model,
for which we provide an explicit formula: the *independent natural extension*.
Our central result is that the independent natural extension satisfies so-called
marginalisation, associativity and *strong factorisation* properties. These
allow us to relate our research to more traditional ways of defining
independence based on factorisation.

Gert de Cooman, Enrique Miranda and Marco Zaffalon. Factorisation properties of the strong product. Proceedings of SMPS'2010. Oviedo (Spain), September 2010.

We investigate a number of factorisation conditions in the framework of sets of probability measures, or coherent lower previsions, with finite referential spaces. We show that the so-called strong product constitutes one way to combine a number of marginal coherent lower previsions into an independent joint lower prevision, and we prove that under some conditions it is the only independent product that satisfies the factorisation conditions.

Matthias Troffaes, Enrique Miranda and Sébastien Destercke. On the connection between probability boxes and possibility measures. Proceedings of EUSFLAT'2011. Aix-les-Bains (France), July 2011.

We explore the relationship between p-boxes on totally preordered spaces and possibility measures. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes.

Gert de Cooman, Enrique Miranda. Independent natural extension for sets of desirable gambles. Proceedings of ISIPTA'2011. Innsbruck (Austria), July 2011.

We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal sets of gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence.

Enrique Miranda, Marco Zaffalon, Gert de
Cooman.
Conglomerable natural extension.
Proceedings of ISIPTA'2011. Innsbruck (Austria), July 2011.

We study the weakest conglomerable model
that is implied by a set of desirability or probabilistic assessments. We call
it the *conglomerable natural extension*. We show that taking the natural
extension of the assessments while imposing conglomerability---which is the
procedure adopted in Walley's theory---does not yield, in general, the
conglomerable natural extension (but it does so in the case of the marginal
extension). Iterating this process originates a sequence of models that approach
the conglomerable natural extension, although it is not known, at this point,
whether or not it is attained in the limit. We give sufficient conditions for
this to happen in some special cases, and study the differences that arise when
we work with desirable gambles or coherent lower previsions. These results seem
to point to the need of re-thinking the foundations of Walley's theory in the
case of infinite spaces of possibility.

Ignacio Montes, Enrique Miranda, Susana Díaz. Imprecise preferences by means of probability boxes. Proceedings of the ERCIM'2011 conference. London (United Kingdom), December 2011.

One of the most popular preference measures between random variables is the notion of stochastic dominance, which is based on the comparison of their cumulative distribution functions. However, it is not uncommon to encounter situations where there is not enough information to fully elicitate the probability distributions of these variables, and it becomes necessary to look for robust alternatives. In this work, we generalise the notion of stochastic dominance to the case where there is uncertainty about the probability distribution of the random variables. The resulting notion entails the comparison of two sets of cumulative distribution functions, which can be equivalently be represented by means of probability boxes, or p-boxes. We propose a number of possible extensions of stochastic dominance, and investigate the relationships between them. In particular, we determine how are the preferences derived from these extensions are affected by considering convex combinations, point-wise limits, and the difference between considering the finitely additive or countably additive subjacent probability distributions. The particular case of 0-1 valued p-boxes, which are the extreme points of the class of probability boxes, is also investigated.

Gert de Cooman, Enrique Miranda. Lower previsions induced by filter maps. Proceedings of IPMU'2012. Catania (Italy), July 2012.

Coherent lower previsions are amongst the
most prominent uncertainty models within imprecise probability theory. We show
how they can model information about the joint behaviour of two variables, when
these are related by means of a *filter map*: a model for the imprecise
observation of a random variable by means of a class of filters. Our
construction preserves a number of interesting properties, such as $n$-monotonicity,
and it generalises a number of existing results for multi-valued mappings.

Enrique Miranda, Marco Zaffalon. Conglomerable coherent lower previsions. Proceedings of SMPS'2012.Konstanz (Germany), October 2012.

Walley's theory of coherent lower
previsions builds upon the former theory by Williams with the explicit aim to
make it deal with conglomerability. We show that such a construction has been
only partly successful because Walley's founding axiom of joint coherence does
not entirely capture the implications of conglomerability. As a way to fully
achieve Walley's original aim, we propose then the new theory of *
conglomerable coherent lower previsions*. We show that Walley's theory
coincides with ours when all conditioning events have positive lower probability,
or when conditioning partitions are nested.

Enrique Miranda, Alejandro García Huergo. Stochastic dominance with non-additive measures. Proceedings of ERCIM'2012. Oviedo (Spain), December 2012.

In decision problems, it is not uncommon to
encounter situations of imprecise knowledge about the probabilities of the
states of nature associated to the different alternatives. One possible model
for this situation is to consider sets of probability measures, which in turn
can be equivalently represented by different types of non-additive measures. The
aim is to study a number of extensions of the criterion of stochastic dominance
to the case where, instead of single probability measures, the credal sets
associated to two non-additive measures are to be compared. The cases of
coherent lower probabilities, 2-monotone capacities, belief functions and
possibility measures are considered. It is shown that the different extensions
of stochastic dominance are able to encompass different attitudes towards risk.
Moreover, the results can be used to compare random sets by means of stochastic
dominance, providing thus a model for dealing with imprecision on the utilities.
The results are illustrated by means of Ellsberg’s paradox and a number of
variations.

Ignacio Montes, Enrique Miranda, Susana Montes. Imprecise statistical preferences. Proceedings of ERCIM'2012. Oviedo (Spain), December 2012.

In decision making under uncertainty, it is not uncommon to encounter situations with vague or conflicting information about the probability distributions of the rewards associated to the different alternatives. In such situations, the elicitation of a unique probability model for each of the alternatives may be difficult, and its use questionable. One of the solutions that have been proposed for situations like this is to model our uncertainty about these probabilities by means of imprecise probability models. Here, two different optimality criteria have been considered: stochastic dominance and statistical preference. The former is based on the comparison of the associated distribution functions, while the latter uses a probabilistic relation and can be seen as a robust alternative to expected utility. These two methods are applied to elicit the preferences between sets of random variables instead of single ones. Some characterizations are provided, and a connection between imprecise statistical preference and the aggregation functions used in voting theory is presented. Finally, a particular case is studied: that of the comparison of random sets. It is proven that under some conditions these methods can be characterized by means of the Choquet integral.

Enrique Miranda, Sebastien Destercke. Extreme points of the credal sets generated by elementary comparative probabilities. Proceedings of ECSQARU'2013. Utrecth (The Netherlands), July 2013.

When using convex probability sets (or, equivalently, lower previsions) as models of uncertainty, identifying extreme points can be useful to perform various computations or to use some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the elements of a finite space. We use these extreme points to study the properties of the lower probability induced by this set, and connect comparative probabilities with other uncertainty models.

Enrique Miranda, Ignacio Montes. Coherent updating of 2-monotone previsions. Proceddings of ISIPTA'2013. Compiègne (France), July 2013.

The conditions for a 2-monotone lower
prevision to be uniquely updated to a conditional lower prevision are determined.
Then a number of particular cases are investigated: completely monotone lower
previsions, for which equivalent conditions in terms of the focal elements of
the associated belief function are established;

random sets, for which some conditions in terms of the measurable selections can
be given; and minitive lower previsions, that are shown to correspond to the
particular case of vacuous lower previsions.

Enrique Miranda, Marco Zaffalon. Computing the conglomerable natural extension. Proceedings of ISIPTA'2013. Compiègne (France), July 2013.

Given a coherent lower prevision __P__,
we consider the problem of computing the smallest coherent lower prevision __F__
≥ __P__ that is conglomerable, in case it exists. __F__ is called the
conglomerable natural extension. Past work has shown that __F__ can be
approximated by an increasing sequence __E__n (n in N) of coherent lower
previsions. We close an open problem by showing that this sequence can be
infinite, while being made of distinct elements. Moreover, we give sufficient
conditions, of quite broad applicability, to make sure that the point-wise limit
of the sequence is __F__ in case __P__ is the lower envelope of finitely
many linear previsions. In addition, we study the question of the existence of
__F__ and its relationship with the notion of marginal extension.

Renato Pelessoni, Paolo Vicig, Ignacio Montes and Enrique Miranda. Imprecise copulas and bivariate stochastic orders. Proceedings of EUROFUSE'2013. Oviedo (Spain), December 2013.

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 pboxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar’s theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

Ignacio Montes, Enrique Miranda and Susana Montes. Connecting interval-valued fuzzy sets with imprecise probabilities. Proceedings of SMPS'2014. Warsaw (Poland), September 2014.

We study interval-valued fuzzy sets as a model for the imprecise knowledge of the membership function of a fuzzy set. We compare three models for the probabilistic information about this membership function: the set of distributions of the measurable selections, the upper and lower probabilities of the associated random interval, and its p-box. We give sufficient conditions for the equality between these sets, and establish a connection with the notion of probability induced by an intuitionistic fuzzy set. An alternative approach to the problem by means of sets of finitely additive distributions is also considered.

Ignacio Montes, Enrique Miranda and Susana Montes. Stochastic orders fo fuzzy random variables. Proceedings of SMPS'2014. Warsaw (Poland), September 2014.

The comparison of random variables can be made by means of stochastic orders such as expected utility or statistical preference. One possible model when the random variables are imprecisely observed is to consider fuzzy random variables, so that the images become fuzzy sets. This paper proposes two comparison methods for fuzzy random variables: one based on fuzzy rankings and another one that uses the extensions of stochastic orders to an imprecise framework. The particular case where the images of the fuzzy random variables are triangular fuzzy numbers is investigated. We illustrate our results by means of a decision making problem.

Ignacio Montes, Enrique Miranda and Susana Montes. Ranking fuzzy sets and fuzzy random variables by means of stochastic orders. Proceedings of IFSA-EUSFLAT'2015. Gijón (Spain), July 2015.

This paper establishes a theory of decision making under uncertainty with fuzzy utilities. The extension of expected utility and stochastic dominance to the comparison of sets of random variables plays a crucial role. Their properties as fuzzy rankings are studied, and their definitions are further generalized to the comparison of fuzzy random variables. Also, a connection between expected utility for fuzzy random variables and the comparison of the lower/upper probabilities they induce is proven.

Enrique Miranda and Marco Zaffalon. Conformity and independence for coherent lower previsions. Proceedings of ISIPTA'2015. Pescara (Italy), July 2015.

We study the conformity of marginal unconditional and
conditional models with a joint model under assumptions of epistemic irrelevance
and independence, within Walley's theory of coherent lower previsions. By doing
so, we make a link with a number of prominent models within this theory: the
marginal extension, the irrelevant natural extension, the independent natural
extension and the strong product.

Arthur van Camp, Gert de Cooman, Enrique Miranda and Erik Quaeghebeur. Modelling indifference with choice functions. Proceedings of ISIPTA'2015. Pescara (Italy), July 2015.

We investigate how to model indifference with choice functions. We take the coherence axioms for choice functions proposed by Seidenfeld, Schervisch and Kadane as a source of inspiration, but modify them to strengthen the connection with desirability. We discuss the properties of choice functions that are coherent under our modified set of axioms and the connection with desirability. Once this is in place, we present an axiomatisation of indifference in terms of desirability. On this we build our characterisation of indifference in terms of choice functions.

Ignacio Montes, Enrique Miranda. Bivariate p-boxes and maxitive functions. Proceedings of IPMU'2016. Eindhoven (The Netherlands), June 2016.

We investigate the properties of the upper probability associated with a bivariate p-box, that may be used as a model for the imprecise knowledge of a bivariate distribution function. We give necessary and su cient conditions for this upper probability to be maxitive, characterize its focal elements, and study which maxitive functions can be obtained as upper probabilities of bivariate p-boxes.

Enrique Miranda, Marco Zaffalon. Full conglomerability, continuity and marginal extension. Proceedings of SMPS'2016. Rome (Italy), September 2016.

We investigate fully conglomerable coherent lower previsions
in the sense of Walley, and some particular cases of interest: envelopes of
fully conglomerable linear previsions, envelopes of countably additive linear
previsions and fully disintegrable linear previsions. We study the connections
with continuity and countable super-additivity, and show that full
conglomerability can be characterised in terms of a supremum of marginal
extension models.

Arthur van Camp, Enrique Miranda, Marco Zaffalon. Lexicographic choice functions without archimedeanicity. Proceedings of SMPS'2016. Rome (Italy), September 2016.

We investigate the connection between choice functions and lexicographic probabilities, by means of the convexity axiom considered by Seidenfeld et al. (2010) but without imposing any Archimedean condition. We show that lexicographic probabilities are related to a particular type of sets of desirable gambles, and investigate the properties of the coherent choice function this induces via maximality. Finally, we show that the convexity axiom is necessary but not sufficient for a coherent choice function to be the infimum of a class of lexicographic ones.

I. Montes, E. Miranda, S. Destercke. A study of the pari-mutuel model from the point of view of imprecise probabilities. Proceedings of ISIPTA'2017. Lugano (Switzerland), July 2017.

The Pari-Mutuel model is a distortion model that has its origin in horse racing. Since then, it has been applied in many different fields, such as finance or risk analysis. In this paper we investigate the properties of the Pari-Mutuel model within the framework of Imprecise Probabilities. Since a Pari-Mutuel model induces (2-monotone) coherent lower and upper probabilities, we investigate its connections with other relevant models within this theory, such as probability intervals and belief functions. We also determine the number of extreme points of the credal set induced by the Pari-Mutuel model and study how to combine the information given by multiple Pari-Mutuel models.

E. Miranda, I. Montes. Game solutions, probability transformations and the core. Proceedings of ISIPTA'2017. Lugano (Switzerland), July 2017.

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations of lower probabilities. The first one coincides with the pignistic transformation proposed in the Transferable Belief Model; the second one is not efficient in general, leading us to propose a normalized version. We consider a number of particular cases of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distorsion models. For them, we provide some alternative expressions of the transformations and study when they belong to the core of the lower probability.

Arthur Van Camp, Enrique Miranda, Gert de Cooman. Natural extension of choice functions. Proceedings of IPMU'2018. Cádiz (Spain), June 2018.

We extend the notion of natural extension, that gives the least committal extension of a given assessment, from the theory of sets of desirable gambles to that of choice functions. We give an expression of this natural extension and characterise its existence by means of a property called avoiding complete rejection. We prove that our notion reduces indeed to the standard one in the case of choice functions determined by binary comparisons, and that these are not general enough to determine all coherent choice function. Finally, we investigate the compatibility of the notion of natural extension with the structural assessment of indifference between a set of options.

Ignacio Montes, Enrique Miranda, Paolo Vicig. Approximations of coherent lower probabilities by 2-monotone capacities. Proceedings of IPMU'2018. Cádiz (Spain), June 2018.

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. While our approach is based on a distance proposed by Baroni and Vicig, we also discuss a number of alternatives. Finally, we show that our work applies to the more general problem of approximating coherent lower previsions.

Ignacio Montes, Enrique Miranda, Paolo Vicig.
Outer approximations of coherent lower probabilities using belief functions.
Proceedings of BELIEF'2018. Compiègne (France), September 2018.
**Best paper award**.

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 in that case we can determine all the undominated outer approximations in a simple manner.

Enrique Miranda, Marco Zaffalon. Compatibility, coherence and the RIP. Proceedings of SMPS'2018. Compiègne (France), September 2018.

We generalise the classical result on the compatibility of marginal, possible non-disjoint, assessments in terms of the running intersection property to the imprecise case, where our beliefs are modelled in terms of sets of desirable gambles. We consider the case where we have unconditional and conditional assessments, and show that the problem can be simplified via a tree decomposition.

Enrique Miranda, Ignacio Montes, Paolo Vicig.* *On
the elicitation of outer approximations of a coherent lower probability*. *
Proceedings of EURO'2019. Dublin (Ireland), June 2019.

When
modelling imprecise or vague information about the outcome of an experiment, one
possibility is to consider a set of probability distributions instead of
electing a single one without sufficient guarantees. Such a set may be
equivalently represented as a coherent lower prevision, by means of lower
envelopes. While this representation is computationally simpler, it still has a
number of drawbacks: for instance, the number of extreme points of the set of
probability distributions may be infinite, and these may be difficult to
characterise.

Enrique Miranda, Ignacio Montes, Sébastien Destercke.
A unifying frame for neighbourhood and distortion models. Proceedings of ISIPTA'2019.
Ghent (Belgium), July 2019. **Best poster award**.

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari-mutuel model, linear vacuous mixtures or the constant odds ratio model. In this paper, we show that all of them can be represented as probability sets that are neighbourhoods defined over different (pre)-metrics, providing a unified view of such models. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, . . . thus providing possible guidelines to pick a neighbourhood rather than another.

Arthur Van Camp, Enrique Miranda. Irrelevant natural extension for choice functions. Proceedings of ISIPTA'2019. Ghent (Belgium), July 2019.

We consider coherent choice functions under the recent axiomatisation proposed by De Bock and de Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment.

Arianna Casanova, Enrique Miranda, Marco Zaffalon. Social pooling of beliefs and values with desirability. Proceedings of FLAIRS'2020.

The problem of aggregating beliefs and values of rational subjects is treated with the formalism of sets of desirable gambles. This leads on the one hand to a new perspective of traditional results of social choice (in particular Arrow’s theorem as well as sufficient conditions for the existence of an oligarchy and democracy) and on the other hand to use the same framework to create connections with opinion pooling. In particular, we show that weak Pareto can be derived as a coherence requirement and discuss the aggregation of state independent beliefs.

Enrique Miranda, Ignacio Montes, Paolo Vicig. On the elicitation of an optimal outer approximation of a coherent lower probability. Proceedings of IPMU'2020. Lisbon (Portugal), June 2020.

The process of outer approximating a coherent lower probability by a more tractable model with additional properties, such as 2- or complete monotonicity, 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 respect to the initial model, or maximising the specificity of the outer approximation. We apply these on 2- and complete monotone lower probabilities, and also on possibility measures.

Alexander Erreygers, Enrique Miranda. A study of the set of probability measures compatible with probability judgements. Proceedings of IPMU'2020. Lisbon (Portugal), June 2020.

We consider a set of comparative probability judgements over a finite possibility space and study the structure of the set of probability measures that are compatible with them. We relate the existence of some compatible probability measure to Walley’s behavioural theory of imprecise probabilities, and introduce a graphical representation that allows us to bound, and in some cases determine, the extreme points of the set of compatible measures. In doing this, we generalise some earlier work by Miranda and Destercke on elementary comparisons.

Enrique Miranda, Ignacio Montes, Sébastien Destercke. Processing distortion models: a comparative study. Proceedings of CMStatistics'2020.

When dealing with sets of probabilities, distortion or neighbourhood models are convenient, practical tools, as very few parameters determine them: an initial probability distribution P0, a distortion factor \delta>0 and a specific distortion procedure. The different choices have led to several different families of neighbourhood models, with applications in robust statistics or machine learning. We compare the performance of several distortion models under several processing procedures. First of all, we study their behaviour when merging different distortion models quantifying uncertainty on the same quantity using conjunction, disjunction or convex mixtures. Secondly, we investigate whether the marginal credal sets of a distortion model are also members of the same family, as well as the procedure for determining a global model from marginal ones using independence or natural extension. The analysis is made for six different families of distortion models: the pari-mutuel, epsilon-contamination, constant odds-ratio, total variation, Kolmogorov and L1.

Marco Zaffalon, Enrique Miranda. The sure thing. Proceedings of ISIPTA'2021. Granada (Spain), July 2021.

If we prefer action a to b both under an event and under its complement, then we should just prefer a to b. This is Savage’s sure-thing principle. In spite of its intuitive- and simple-looking nature, for which it gets almost immediate acceptance, the sure thing is not a logical principle. So where does it get its support from? In fact, the sure thing may actually fail. This is related to a variety of deep and foundational concepts in causality, decision theory, and probability, as well as to Simpsons’ paradox and Blyth’s game. In this paper we try to systematically clarify such a network of relations. Then we propose a general desirability theory for nonlinear utility scales. We use that to show that the sure thing is primitive to many of the previous concepts: In non-causal settings, the sure thing follows from considerations of temporal coherence and coincides with conglomerability; it can be understood as a rationality axiom to enable well-behaved conditioning in logic. In causal settings, it can be derived using only coherence and a causal independence condition.

Sébastien Destercke, Ignacio Montes, Enrique Miranda. Processing multiple distortion models: a comparative study. Proceedings of ISIPTA'2021. Granada (Spain), July 2021.

When dealing with uncertain information, distortion or neighbourhood models are convenient practical tools, as they rely on very few parameters. In this paper, we study their behaviour when such models are combined and processed. More specifically, we study their behaviour when merging different distortion models quantifying uncertainty on the same quantity, and when manipulating distortion models defined over multiple variables.

Enrique Miranda, Ignacio
Montes.* *Centroids of credal sets: a comparative
study. Proceedings of ECSQARU'2021. Prague (Czech Republic), September 2021.

We compare a number of different notions of centroid of a credal set: the Shapley value, that arises in the context of game theory; the average of the extreme points; the incenter with respect to the total variation distance between probability measures; and the limit of a procedure of uniform contraction. We show that these four centers do not coincide in general, give some sufficient conditions for their equality, and analyse their axiomatic properties. Finally, we discuss briefly how to define a notion of centrality measure.