# JOURNAL PUBLICATIONS

The available full papers constitute earlier versions of the published ones

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol 10 (1), pp. 1--15, 2002

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RELATIONSHIPS BETWEEN POSSIBILITY MEASURES AND NESTED RANDOM SETS

Different authors have observed some relationships between consonant random sets and possibility measures, specially for finite universes. In this paper, we go deeply into this matter and propose several possible definitions for the concept of consonant random set. Three of these conditions are equivalent for finite universes. In that case, the random set considered is associated to a possibility measure if and only if any of them is satisfied. However, in a general context, none of the six definitions here proposed is sufficient for a random set to induce a possibility measure. Moreover, only one of them seems to be necessary.

International Journal of Approximate Reasoning, 32(1), pp. 23--42, 2003.

## ENRIQUE MIRANDA, GERT DE COOMAN. EPISTEMIC INDEPENDENCE IN NUMERICAL POSSIBILITY THEORY

Numerical possibility measures can be interpreted as systems of upper betting rates for events. As such, they have a special part in the unifying behavioural theory of imprecise probabilities, proposed by Walley. On this interpretation, they should arguably satisfy certain rationality, or consistency, requirements, such as avoiding sure loss and coherence. Using a version of Walley's notion of epistemic independence suitable for possibility measures, we study in detail what these rationality requirements tell us about the construction of independent product possibility measures from given marginals, and we obtain necessary and sufficient conditions for a product to satisfy these criteria. In particular, we show that the well-known minimum and product rules for forming independent joint distributions from marginal ones, are only coherent when at least one of these distributions assume just the values zero and one.

International Journal of Approximate Reasoning, Vol 33(1), pp. 95--115, 2003.

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, EXTREME POINTS OF CREDAL SETS GENERATED BY 2-ALTERNATING CAPACITIES

The characterization of the extreme points constitutes a crucial issue in the investigation of convex sets of probabilities, not only from a purely theoretical point of view, but also as a tool in the management of imprecise information. In this respect, different authors have found an interesting relation between the extreme points of the class of probability measures dominated by a second order alternating Choquet capacity and the permutations of the elements in the referential. However, they have all restricted their work to the case of a finite referential space. In an infinite setting, some technical complications arise and they have to be carefully treated. In this paper, we extend the mentioned result to the more general case of separable metric spaces. Furthermore, we derive some interesting topological properties about the convex sets of probabilities here investigated. Finally, a closer look to the case of possibility measures is given: for them, we prove that the number of extreme points can be reduced even in the finite case.

Information Sciences, Vol 168(1--4), pp. 51--75, 2004

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, A RANDOM SET CHARACTERISATION OF POSSIBILITY MEASURES

Several authors have pointed out the relationship between consonant random sets and possibility measures. However, this relationship has only been proven for the finite case, where the inverse Möbius of the upper probability induced by the random set simplifies the computations to a great extent. In this paper, we study the connection between both concepts for arbitrary referential spaces. We complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases.

Journal of Statistical Planning and Inference, Vol 133(1), pp. 173-197, 2005.

## ENRIQUE MIRANDA, GERT DE COOMAN AND INÉS COUSO, LOWER PREVISIONS INDUCED BY MULTI-VALUED MAPPINGS

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 their explanatory and unifying power.

Journal of Mathematical Analysis and Applications, Vol 307(1), pp. 32-47, 2005

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RANDOM SETS AS IMPRECISE RANDOM VARIABLES

Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the probability 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 probabilities induced by the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.

Fuzzy Sets and Systems, Vol 154(3), pp. 386-412, 2005.

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RANDOM INTERVALS AS A MODEL FOR IMPRECISE INFORMATION

Random intervals constitute one of the classes of random sets with a greater number of applications. In this paper, we regard them as the imprecise observation of a random variable, and study how to model the information about the probability distribution of this one. Two possible models are the probability distributions of the measurable selections and those bounded by the upper probability. We prove that, under some hypotheses, the closures of these two sets in the topology of the weak convergence coincide, improving results from the literature. Moreover, we provide examples showing that the two models are not equivalent in general, and give sufficient conditions for the equality between them. Finally, we comment on the relationship between random intervals and fuzzy numbers.

Journal of Intelligent and Fuzzy Systems, Vol 16(4) (Special Issue dedicated to the 60th birthday of Etienne E. Kerre), pp. 253-263, 2005.

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA. N-MONOTONE LOWER PREVISIONS

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.

Journal of Theoretical Probability, Vol 20(3), 663-693, 2007

## ENRIQUE MIRANDA, GERT DE COOMAN AND ERIK QUAEGHEBEUR, THE HAUSDORFF MOMENT PROBLEM UNDER FINITE ADDITIVITY

We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.

International Journal of Approximate Reasoning, Vol 46(1), 188-225, 2007

## ENRIQUE MIRANDA AND GERT DE COOMAN, MARGINAL EXTENSION IN THE THEORY OF COHERENT LOWER PREVISIONS

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

Journal of Mathematical Analysis and Applications, 340 (2), 982-999, 2008

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA, A UNIFYING APPROACH TO INTEGRATION FOR BOUNDED POSITIVE CHARGES

This paper deals with n-monotone functionals, which constitute a generalisation of n-monotone set functions. Using the notion of exactness of a functional, we introduce a new notion of lower and upper integral which subsumes as particular cases most of the approaches to integration in the literature. As a consequence, we can characterise which types of integrals can be used to calculate the natural extension (the lower envelope of all linear extensions) of a positive bounded charge.

International Journal of Approximate Reasoning, 48(1), 132-155, 2008

## ENRIQUE MIRANDA, GERT DE COOMAN AND ERIK QUAEGHEBEUR, FINITELY ADDITIVE EXTENSIONS OF DISTRIBUTION FUNCTIONS AND MOMENT SEQUENCES: THE COHERENT LOWER PREVISION APPROACH

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions.

International Journal of Approximate Reasoning, 48(2), 628-658, 2008

## ENRIQUE MIRANDA, A SURVEY OF THE THEORY OF COHERENT LOWER PREVISIONS.

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.

Journal of Statistical Planning and Inference, 138(8), 2409-2432, 2008

## GERT DE COOMAN AND ENRIQUE MIRANDA, WEAK AND STRONG LAWS OF LARGE NUMBERS FOR COHERENT LOWER PREVISIONS

We prove weak and strong laws of large numbers for coherent lower previsions, where the lower prevision of a random variable is given a behavioural interpretation as a subject's supremum acceptable price for buying it. Our laws are a consequence of the rationality criterion of coherence, and they can be proven under assumptions that are surprisingly weak when compared to the standard formulation of the laws in more classical approaches to probability theory.

Journal of Mathematical Analysis and Applications, 347(1), 143--156, 2008

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA. N-MONOTONE EXACT FUNCTIONALS

We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals.

Artificial Intelligence, 173(1), 104--144, 2009

## ENRIQUE MIRANDA AND MARCO ZAFFALON. COHERENCE GRAPHS

We study the consistency of a number of probability distributions, which are allowed to be imprecise. To make the treatment as general as possible, we represent those probabilistic assessments as a collection of conditional lower previsions. The problem then becomes proving Walley's (strong) coherence of the assessments. In order to maintain generality in the analysis, we assume to be given nearly no information about the numbers that make up the lower previsions in the collection. Under this condition, we investigate the extent to which the above global task can be decomposed into simpler and more local ones. This is done by introducing a graphical representation of the conditional lower previsions that we call the coherence graph: we show that the coherence graph allows one to isolate some subsets of the collection whose coherence is sufficient for the coherence of all the assessments; and we provide a polynomial-time algorithm that finds the subsets efficiently. We show some of the implications of our results by focusing on three models and problems: Bayesian and credal networks, of which we prove coherence; the compatibility problem, for which we provide an optimal graphical decomposition; probabilistic satisfiability, of which we show that some intractable instances can instead be solved efficiently by exploiting coherence graphs

Journal of Statistical Planning and Inference, 139(2), 256-276, 2009

## GERT DE COOMAN AND ENRIQUE MIRANDA, FORWARD IRRELEVANCE

We investigate how to combine marginal assessments about the values that random variables assume separately into a model for the values that they assume jointly, when (i) these marginal assessments are modelled by means of coherent lower previsions, and (ii) we have the additional assumption that the random variables are forward epistemically irrelevant to each other. We consider and provide arguments for two possible combinations, namely the forward irrelevant natural extension and the forward irrelevant product, and we study the relationships between them. 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

International Journal of Approximate Reasoning, 50(2), special issue on the Imprecise Dirichlet Model, 204-216, 2009

## GERT DE COOMAN, ENRIQUE MIRANDA AND ERIK QUAEGHEBEUR, REPRESENTATION INSENSITIVITY IN IMMEDIATE PREDICTION

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. We establish that such representation insensitive predictive models have very interesting properties, and show that among such models, the ones produced by the Imprecise Dirichlet (Multinomial) Model are quite special in a number of ways. In the Conclusion, we discuss the open question as to how unique the Imprecise Dirichlet (Multinomial) Model predictive lower previsions are in being representation insensitive.

Fuzzy Sets and Systems, 160(9), 1286-1307, 2009

## ENRIQUE MIRANDA, UPDATING COHERENT PREVISIONS ON FINITE SPACES

We compare the different notions of conditional 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, and we discuss the extensions of our results to infinite spaces.

Journal of Artificial Intelligence Research, 34, 757-821, 2009

## MARCO ZAFFALON AND ENRIQUE MIRANDA. CONSERVATIVE INFERENCE RULE FOR UNCERTAIN REASONING UNDER INCOMPLETENESS

In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process' behaviour to be partly unknown. Then we use Walley's theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classi cation, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences.

Bernoulli, 15(3), 721-735, 2009

## GERT DE COOMAN, ERIK QUAEGHEBEUR AND ENRIQUE MIRANDA, EXCHANGEABLE LOWER PREVISIONS

We extend de Finetti's notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We derive representation theorems in both the finite and the countable case, in terms of sampling without and with replacement, respectively.

Information Sciences, 180(8), 1407-1417, 2010

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL. APPROXIMATION OF UPPER AND LOWER PROBABILITIES BY MEASURABLE SELECTIONS

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. In doing this, we generalise a number of results from the literature. Finally, we study the particular case of consonant random sets and we also derive a relationship between Aumann and Choquet integrals.

Journal of Statistical Planning and Inference, 140(7), 1805-1833, 2010

## ENRIQUE MIRANDA AND MARCO ZAFFALON, CONDITIONAL MODELS: COHERENCE AND INFERENCE THROUGH SEQUENCES OF JOINT MASS FUNCTIONS

We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus on two problems: (coherence) giving conditions to guarantee that a conditional model is self-consistent; (inference) delivering methods to derive new probabilistic statements from a self-consistent conditional model. We address these problems in the case where the probabilistic statements can be specified imprecisely through sets of probabilities, while restricting the attention to finite spaces of possibilities. Using Walley's theory of coherent lower previsions, we fully characterise the question of coherence, and specialise it for the case of precisely specified probabilities, which is the most common case addressed in the literature. This shows that coherent conditional models are equivalent to sequences of (possibly sets of) unconditional mass functions. In turn, this implies that the inferences from a conditional model are the limits of the conditional inferences obtained by applying Bayes' rule, when possible, to the elements of the sequence. In doing so, we unveil the tight connection between conditional models and zero-probability events. Such a connection appears to have been overlooked by most previous works on the subject, thus preventing so far to give a full account of coherence and inference for conditional models.

Annals of Mathematics and Artificial Intelligence, 60(3-4), 251-309, 2010

## ENRIQUE MIRANDA AND MARCO ZAFFALON. NOTES ON DESIRABILITY AND CONDITIONAL LOWER PREVISIONS

We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley's celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional lower previsions is derived from a set of desirable gambles, and its converse. In either case, we relate the properties of the derived model with those of the originating one. Our results constitute basic tools to move from one formalism to the other, and thus to take advantage of work done in the two fronts. Sets of desirable gambles are at the same time very powerful and intuitive models of uncertainty. Given the central role of uncertainty in artificial intelligence, this work marks a key passage towards the wider accessibility of those modelling capabilities in artificial intelligence.

IEEE Transactions on Automatic Control, 56(7), 1567--1581, 2011

## ALESSIO BENAVOLI, MARCO ZAFFALON AND ENRIQUE MIRANDA, A NEW ROBUST APPROACH TO FILTERING BASED ON COHERENT LOWER PREVISIONS

The classical filtering problem is re-examined to take into account imprecision in the knowledge about the probabilistic relationships involved. To achieve that, we consider closed convex 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.

Artificial Intelligence, 175(12-13), 1911--1950, 2011

## GERT DE COOMAN, ENRIQUE MIRANDA AND MARCO ZAFFALON. INDEPENDENT NATURAL EXTENSION

There is no unique extension of the standard notion of probabilistic independence to the case where probabilities are indeterminate or imprecisely specified. Epistemic independence is an extension that formalises the intuitive idea of mutual irrelevance between different sources of information. It has a wide scope and great appeal, especially for a field like Artificial Intelligence, where such an idea or interpretation of independence has already been employed quite often in precise-probabilistic contexts. Despite of this, epistemic independence has received little attention so far. This paper develops its foundations for variables assuming values in finite spaces. We define (epistemically) independent products of marginals (or possibly conditionals) and show that there always is a unique least-committal such independent product, which we call the independent natural extension.We supply an explicit formula for it, and study some of its properties: associativity, marginalisation, external additivity, which are basic tools to work with the independent natural extension. Additionally, we consider a number of ways in which the standard factorisation formula for independence can be generalised to an imprecise-probabilistic context. We show, under some mild conditions, that when the focus is on least-committal models, using the independent natural extension is equivalent to imposing a so-called strong factorisation property. This is an important outcome for applications as it gives a simple tool to make sure that inferences are consistent with epistemic independence judgments. We discuss the potential of our results for applications in Artificial Intelligence by recalling recent work by some of us, where the independent natural extension was applied to graphical models. It has allowed, for the first time, the development of an exact linear-time algorithm for the imprecise-probability updating of credal trees.

International Journal of Approximate Reasoning, 53(8), 1200--1227, 2012

## ENRIQUE MIRANDA, MARCO ZAFFALON AND GERT DE COOMAN. CONGLOMERABLE NATURAL EXTENSION

At the foundations of probability theory lies a question that has been open since de Finetti framed it in 1930: whether or not an uncertainty model should be required to be conglomerable. Conglomerability is related to accepting infinitely many conditional bets. Walley is one of the authors who have argued in favor of conglomerability, while de Finetti rejected the idea. In this paper we study the extension of the conglomerability condition to two types of uncertainty models that are more general than the ones envisaged by de Finetti: sets of desirable gambles and coherent lower previsions. We focus in particular on the weakest (i.e., the least-committal) of those extensions, which we call the conglomerable natural extension. The weakest extension that does not take conglomerability into account is simply called the natural extension. We show that taking the natural extension of assessments after imposing conglomerability---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 of imposing conglomerability and taking the natural extension produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether this sequence converges to it. We give sufficient conditions for this to happen in some special cases, and study the differences between working with coherent sets of desirable gambles and coherent lower previsions. Our results indicate that it is necessary to rethink the foundations of Walley's theory of coherent lower previsions for infinite partitions of conditioning events

Journal of Artificial Intelligence Research, 45, 601--640, 2012

## GERT DE COOMAN AND ENRIQUE MIRANDA. IRRELEVANT AND INDEPENDENT NATURAL EXTENSION FOR SETS OF DESIRABLE GAMBLES

The results in this paper add useful tools to the theory of sets of desirable gambles, a growing toolbox for reasoning with partial probability assessments. 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 coherent sets of desirable 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. Having such a set of tools that are easily implemented in computer programs is clearly beneficial to fields, like AI, with a clear interest in coherent reasoning under uncertainty using general and robust uncertainty models that require no full specification

Information Sciences, 224, 88--108, 2013

## MATTHIAS TROFFAES, ENRIQUE MIRANDA AND SÉBASTIEN DESTERCKE. ON THE CONNECTION BETWEEN PROBABILITY BOXES AND POSSIBILITY MEASURES

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

Artificial Intelligence, 198, 1--51, 2013

## MARCO ZAFFALON AND ENRIQUE MIRANDA. PROBABILITY AND TIME

Probabilistic reasoning is often attributed a temporal meaning, in which conditioning is regarded as a normative rule to compute future beliefs out of current beliefs and observations. However, the well-established ‘updating interpretation’of conditioning is not concerned with beliefs that evolve in time, and in particular with future beliefs. On the other hand, a temporal justification of conditioning was proposed already by De Moivre and Bayes, by requiring that current and future beliefs be consistent. We reconsider the latter proposal while dealing with a generalised version of the problem, using a behavioural theory of imprecise probability in the form of coherent lower previsions as well as of coherent sets of desirable gambles, and letting the possibility space be finite or infinite. We obtain that using conditioning is normative, in the imprecise case, only if one establishes future behavioural commitments at the same time of current beliefs. In this case it is also normative that present beliefs be conglomerable, which is a result that touches on a long-term controversy at the foundations of probability. In the remaining case, where one commits to some future behaviour after establishing present beliefs, we characterise the several possibilities to define consistent future assessments; this shows in particular that temporal consistency does not preclude changes of mind. And yet, our analysis does not support that rationality requires consistency in general, even though pursuing consistency makes sense and is useful, at least as a way to guide and evaluate the assessment process. These considerations narrow down in the special case of precise probability, because this formalism cannot distinguish the two different situations illustrated above: it turns out that the only consistent rule is conditioning and moreover that it is not rational to be willing to stick to precise probability while using a rule different from conditioning to compute future beliefs; rationality requires in addition the disintegrability of the present-time probability.

International Journal of Approximate Reasoning, 54(9), 1322--1350, 2013

## ENRIQUE MIRANDA AND MARCO ZAFFALON. CONGLOMERABLE COHERENCE

We contrast Williams' and Walley's theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them the most: Williams' theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walley's notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considered also some previous results in the literature, we deduce that Williams' theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walley's theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.

Computational Statistics and Data Analysis, 71(C), 867-885, 2014

## IGNACIO MONTES, ENRIQUE MIRANDA AND SUSANA MONTES. STOCHASTIC DOMINANCE WITH IMPRECISE INFORMATION

Stochastic dominance, which is based on the comparison of distribution functions, is one of the most popular preference measures. However, its use is limited to the case where the goal is to compare pairs of distribution functions, whereas in many cases it is interesting to compare sets of distribution functions: this may be the case for instance when the available information does not allow to fully elicitate the probability distributions of the random variables. To deal with these situations, a number of generalisations of the notion of stochastic dominance are proposed; their connection with an equivalent p-box representation of the sets of distribution functions is studied; a number of particular cases, such as sets of distributions associated to possibility measures, are investigated; and an application to the comparison of the Lorenz curves of countries within the same region is presented

Journal of Mathematical Analysis and Applications, 410(1), 101--116, 2014

## GERT DE COOMAN AND ENRIQUE MIRANDA. LOWER PREVISIONS INDUCED BY FILTER MAPS

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.

European Journal of Operational Research, 234(1), 209--220, 2014

## IGNACIO MONTES, ENRIQUE MIRANDA AND SUSANA MONTES. DECISION MAKING WITH IMPRECISE UTILITIES AND BELIEFS BY MEANS OF STATISTICAL PREFERENCE AND STOCHASTIC DOMINANCE

A problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered. A number of choice rules are proposed and their main properties are investigated, focusing particularly on the generalizations of stochastic dominance and statistical preference. The particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered.

International Journal of Approximate Reasoning, 56(A), 1--27, 2015

## ENRIQUE MIRANDA, MARCO ZAFFALON. ON THE PROBLEM OF COMPUTING THE CONGLOMERABLE NATURAL EXTENSION

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension, provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension. In this paper we consider probabilistic assessments in the form of a coherent lower prevision P, which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F ≥ P that is conglomerable, in case it exists. Past work has shown that F can be approximated by an increasing sequence (F n)n of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many 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

International Journal of Approximate Reasoning, 56(B), 159-177, 2015

## ENRIQUE MIRANDA, IGNACIO MONTES. COHERENT UPDATING OF NON-ADDITIVE MEASURES

The conditions under which a 2-monotone lower prevision can 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, which are shown to correspond to the particular case of vacuous lower previsions.

Fuzzy Sets and Systems, 278C, 48-66, 2015

## IGNACIO MONTES, ENRIQUE MIRANDA, RENATO PELESSONI, PAOLO VICIG. SKLAR'S THEOREM IN AN IMPRECISE SETTING

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 single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

Information Sciences, 298, 373-389, 2015

## ENRIQUE MIRANDA, MATTHIAS TROFFAES, SÉBASTIEN DESTERCKE. A GEOMETRIC AND GAME-THEORETIC STUDY OF THE CONJUNCTION OF POSSIBILITY MEASURES

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical ways to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures.

Journal of Mathematical Psychology, 64/65, 44-57, 2015

## ENRIQUE MIRANDA, SÉBASTIEN DESTERCKE. EXTREME POINTS OF THE CREDAL SETS GENERATED BY COMPARATIVE PROBABILITIES

When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixture 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.

Journal of Mathematical Analysis and Applications, 425/1, 460-488, 2015

## ENRIQUE MIRANDA, MARCO ZAFFALON. INDEPENDENT PRODUCTS IN INFINITE SPACES

Probabilistic independence, intended as the mutual irrelevance of given variables, can be solidly founded on a notion of self-consistency of an uncertainty model, in particular when probabilities go imprecise. There is nothing in this approach that prevents it from being adopted in very general setups, and yet it has mostly been detailed for variables taking finitely many values. In this mathematical study, we complement previous research by exploring the extent to which such an approach can be generalised. We focus in particular on the independent products of two variables. We characterise the main notions, including some of factorisation and productivity, in the general case where both spaces can be infinite and show that, however, there are situations---even in the case of precise probability---where no independent product exists. This is not the case as soon as at least one space is finite. We study in depth this case at the frontiers of well-behaviour detailing the relations among the most important notions; we show for instance that being an independent product is equivalent to a certain productivity condition. Then we step back to the general case: we give conditions for the existence of independent products and study ways to get around its inherent limitations.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 24/2, 229-263, 2016

## RENATO PELESSONI, PAOLO VICIG, IGNACIO MONTES, ENRIQUE MIRANDA. BIVARIATE P-BOXES

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) equivalent to 2-coherence. We therefore focus on the relevant subclass of coherent p-boxes, corresponding to coherent lower probabilities on special domains. Several properties of coherent p-boxes are investigated and compared with those of (one-dimensional) p-boxes or of bivariate distribution functions.

International Journal of Approximate Reasoning, 78/C, 125-137, 2016

## ENRIQUE MIRANDA, MARCO ZAFFALON. CONFORMITY AND INDEPENDENCE WITH COHERENT LOWER PREVISIONS

We define the conformity of marginal and conditional models with a joint model within Walley's theory of coherent lower previsions. Loosely speaking, conformity means that the joint can reproduce the marginal and conditional models we started from. By studying conformity with and without additional assumptions of epistemic irrelevance and independence, we establish connections with a number of prominent models in Walley's theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.

Fuzzy Optimization and Decision Making, 16/4, 297-327, 2017

## IGNACIO MONTES, ENRIQUE MIRANDA, SUSANA MONTES. IMPRECISE STOCHASTIC ORDERS AND FUZZY RANKINGS

We extend the notion of stochastic order to the pairwise comparison of fuzzy random variables. We consider expected utility, stochastic dominance and statistical preference, which are related to the comparisons of the expectations, distribution functions and medians of the underlying variables, and discuss how to generalize these notions to the fuzzy case, when an epistemic interpretation is given to the fuzzy random variables. In passing, we investigate to which extent the earlier extensions of stochastic dominance and expected utility to the comparison of sets of random variables can be useful as fuzzy rankings.

Journal of Statistical Theory and Practice, 11/4, 634-669, 2017

## ENRIQUE MIRANDA, MARCO ZAFFALON. FULL CONGLOMERABILITY

We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley in his 1991 book. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, super-additivity and marginal extension. Moreover, we discuss the special case where a model is conglomerable with respect to a subset of all the partitions, and try to sort out the different notions of conglomerability present in the literature. We conclude that countable additivity, which is routinely used to impose full conglomerability in the precise case, appears to be the most well-behaved way to do so in the imprecise case as well by taking envelopes of countably additive probabilities. Moreover, we characterise these envelopes by means of a number of necessary and sufficient conditions.We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley in his 1991 book. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, super-additivity and marginal extension. Moreover, we discuss the special case where a model is conglomerable with respect to a subset of all the partitions, and try to sort out the different notions of conglomerability present in the literature. We conclude that countable additivity, which is routinely used to impose full conglomerability in the precise case, appears to be the most well-behaved way to do so in the imprecise case as well by taking envelopes of countably additive probabilities. Moreover, we characterise these envelopes by means of a number of necessary and sufficient conditions.

International Journal of General Systems, 46/4, 354-385, 2017

## IGNACIO MONTES, ENRIQUE MIRANDA. BIVARIATE P-BOXES AND MAXITIVE FUNCTIONS

We give necessary and sufficient conditions for a maxitive function to be the upper probability of a bivariate p-box, in terms of its associated possibility distribution and its focal sets. This allows us to derive conditions in terms of the lower and upper distribution functions of the bivariate p-box. In particular, we prove that only bivariate p-boxes with a non-informative lower or upper distribution function may induce a maxitive function. In addition, we also investigate the extension of Sklar’s theorem to this context.

Fuzzy Sets and Systems, 341C, 1-36, 2018

## ARTHUR VAN CAMP, GERT DE COOMAN, ENRIQUE MIRANDA, ERIK QUAEGHEBEUR. COHERENT CHOICE FUNCTIONS, DESIRABILITY AND INDIFFERENCE

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 definition of indifference in terms of choice functions, which we discuss in some detail.

International Journal of Approximate Reasoning, 88/C, 387-400, 2017

## PATRIZIA BERTI, ENRIQUE MIRANDA, PIETRO RIGO. BASIC IDEAS UNDERLYING CONGLOMERABILITY AND DISINTEGRABILITY

The basic mathematical theory underlying the notions of conglomerability and disintegrability is reviewed. Both the precise and the imprecise cases are concerned.

Journal of Artificial Intelligence Research, 60, 1057-1126, 2017

## MARCO ZAFFALON, ENRIQUE MIRANDA. AXIOMATISING INCOMPLETE PREFERENCES THROUGH SETS OF DESIRABLE GAMBLES

We establish the equivalence of two very general theories: the first is the decision-theoretic formalisation of incomplete preferences based on the mixture independence axiom; the second is the theory of coherent sets of desirable gambles (bounded variables) developed in the context of imprecise probability and extended here to vector-valued gambles. Such an equivalence allows us to analyse the theory of incomplete preferences from the point of view of desirability. Among other things, this leads us to uncover an unexpected and clarifying relation: that the notion of state independence—the traditional assumption that we can have separate models for beliefs (probabilities) and values (utilities)—coincides with that of strong independence in imprecise probability; this connection leads us also to propose much weaker, and arguably more realistic, notions of state independence. Then we simplify the treatment of complete beliefs and values by putting them on a more equal footing. We study the role of the Archimedean condition—which allows us to actually talk of expected utility—, identify some weaknesses and propose alternatives that solve these. More generally speaking, we show that desirability is a valuable alternative foundation to preferences for decision theory that streamlines and unifies a number of concepts while preserving great generality. In addition, the mentioned equivalence shows for the first time how to extend the theory of desirability to imprecise non-linear utility, thus enabling us to formulate one of the most powerful self-consistent theories of reasoning and decision-making available today.

International Journal of Approximate Reasoning, 92, 97-119, 2018

## ARTHUR VAN CAMP, GERT DE COOMAN, ENRIQUE MIRANDA. LEXICOGRAPHIC CHOICE FUNCTIONS

We investigate a generalisation of the coherent choice functions considered by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no Archimedeanity condition. We define our choice functions on vector spaces of options, which allows us to incorporate as special cases both Seidenfeld et al.’s (2010) choice functions on horse lotteries and also pairwise choice—which is equivalent to sets of desirable gambles (Quaeghebeur, 2014)—, and to investigate their connections. We show that choice functions based on sets of desirable options (gambles) satisfy Seidenfeld’s convexity axiom only for very particular types of sets of desirable options, which are exactly those that are representable by lexicographic probability systems that have no non-trivial Savage-null events. We call them lexicographic choice functions. Finally, we prove that these choice functions can be used to determine the most conservative convex choice function associated with a given binary relation.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26/6, 1017-1051, 2018

## IGNACIO MONTES, ENRIQUE MIRANDA. EXTREME POINTS OF THE CORE OF POSSIBILITY MEASURES AND P-BOXES

Under an epistemic interpretation, an upper probability can be regarded as equivalent to the set of probability measures it dominates, sometimes referred to as its core. In this paper, we study the properties of the number of extreme points of the core of a possibility measure, and investigate in detail those associated with (uni- and bi-)variate p-boxes, that model the imprecise information about a cumulative distribution function.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26/6, 917-947, 2018

## ENRIQUE MIRANDA, IGNACIO MONTES. SHAPLEY AND BANZHAF VALUES AS PROBABILITY TRANSFORMATIONS

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations. 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 consider its normalized version. We study a number of particular models of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distortion models. For them, we provide some alternative expressions of the Shapley and Banzhaf values and study under which conditions they belong to the core of the lower probability.

International Journal of Approximate Reasoning, 101, 181-205, 2018

## IGNACIO MONTES, ENRIQUE MIRANDA, PAOLO VICIG. 2-MONOTONE OUTER APPROXIMATIONS OF COHERENT LOWER PROBABILITIES

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: quadratic programming, extensions of the total variation distance, and the Weber set from game theory. Finally, we show that our work applies to the more general problem of approximating coherent lower previsions.

Information Sciences, 481, 550-573, 2019

## IGNACIO MONTES, ENRIQUE MIRANDA, SÉBASTIEN DESTERCKE. PARI-MUTUEL PROBABILITIES AS AN UNCERTAINTY MODEL

The pari-mutuel model is a betting scheme that has its origins in horse racing, and that has been applied in a number of contexts, mostly economics. In this paper, we consider the set of probability measures compatible with a pari-mutuel model, characterize its extreme points, and investigate the properties of the associated lower and upper probabilities. We show that the pari-mutuel model can be embedded within the theory of probability intervals, and prove necessary and sufficient conditions for it to be a belief function or a minitive measure. In addition, we also investigate the combination of different pari-mutuel models and their definition on product spaces.

International Journal of Approximate Reasoning, 110, 1-30, 2019

## IGNACIO MONTES, ENRIQUE MIRANDA, PAOLO VICIG. OUTER APPROXIMATING COHERENT LOWER PROBABILITIES WITH BELIEF FUNCTIONS

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 the same time as close as possible to the initial coherent lower probability while not including additional information. We show that this problem can be tackled by means of linear programming, and investigate the features of the set of optimal solutions. Moreover, we emphasize the differences with the outer approximations by 2-monotone lower probabilities. We also study the problem for two particular cases of belief functions that are computationally easier to handle: necessity measures and probability boxes.

Artificial Intelligence, 283C, 2020

## ENRIQUE MIRANDA, MARCO ZAFFALON. COMPATIBILITY, DESIRABILITY AND THE RUNNING INTERSECTION PROPERTY

Compatibility is the problem of checking whether some given probabilistic assessments have a common joint probabilistic model. When the assessments are unconditional, the problem is well established in the literature and finds a solution through the running intersection property (RIP). This is not the case of conditional assessments. In this paper, we study the compatibility problem in a very general setting: any possibility space, unrestricted domains, imprecise (and possibly degenerate) probabilities. We extend the unconditional case to our setting, thus generalising most of previous results in the literature. The conditional case turns out to be fundamentally different from the unconditional one. For such a case, we prove that the problem can still be solved in general by (RIP) but in a more involved way: by constructing a junction tree and propagating information over it. Still, (RIP) does not allow us to optimally take advantage of sparsity: in fact, conditional compatibility can be simplified further by joining junction trees with coherence graphs.

International Journal of General Systems, 49/6, 602-635, 2020

## IGNACIO MONTES, ENRIQUE MIRANDA, AND SÉBASTIEN DESTERCKE. UNIFYING NEIGHBOURHOOD AND DISTORTION MODELS. PART I: NEW RESULTS ON OLD MODELS

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, the linear vacuous mixtures or the constant odds ratio model. This paper is the first part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods defined over specific metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This first part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our findings.

The version below includes a small correction in the convexity axiom Ax.4, that does not affect the validity of the results.

International Journal of General Systems, 49/6, 636-674, 2020

## IGNACIO MONTES, ENRIQUE MIRANDA AND SÉBASTIEN DESTERCKE. UNIFYING NEIGHBOURHOOD AND DISTORTION MODELS. PART II: NEW MODELS AND SYNTHESIS

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. In the first part of this study, we introduced a general, unified view encompassing such neighbourhoods, and revisited some well-known models (pari mutuel, linear vacuous, constant odds-ratio). In this second part, we study models that have received little to no attention, but are induced by classical distances between probabilities, such as the total variation, the Kolmogorov and the L 1 distances. We finish by comparing those models in terms of a number of properties: precision, number of extreme points, n-monotonicity, ...thus providing possible guidelines for selecting a neighbourhood rather than another.

International Journal of Approximate Reasoning, 125C, 49-72, 2020

## ARTHUR VAN CAMP, ENRIQUE MIRANDA. MODELLING EPISTEMIC IRRELEVANCE WITH CHOICE FUNCTIONS.

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.

Fuzzy Sets and Systems, 424C, 1-36, 2021

## ENRIQUE MIRANDA, IGNACIO MONTES, PAOLO VICIG. ON THE SELECTION OF AN OPTIMAL OUTER APPROXIMATION OF A COHERENT LOWER PROBABILITY.

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 studied the problem of outer approximating a given coherent lower probability by a more tractable model, such as a 2- or completely monotone lower probability. Unfortunately, such an outer approximation is not unique in general, even if we restrict our attention to those that are undominated by other models from the same family. In this paper, we investigate whether a number of approaches may help in selecting a unique undominated outer approximation. These are based on minimising a distance with respect to the initial model, maximising the specificity, or preserving the same preferences as the original model. We apply them to 2-and completely monotone approximating lower probabilities, and also to the particular cases of possibility measures and p-boxes.

Annals of Mathematics and Artificial Intelligence, 89(10-11), 965-1011, 2021

## ARIANNA CASANOVA, ENRIQUE MIRANDA, MARCO ZAFFALON. JOINT DESIRABILITY FOUNDATIONS OF SOCIAL CHOICE AND OPINION POOLING

We develop joint foundations for the fields of social choice and opinion pooling using coherent sets of desirable gambles, a general uncertainty model that allows to encompass both complete and incomplete preferences. 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 using the same framework to analyse opinion pooling. In particular, we argue that weak Pareto (unanimity) should be given the status of a rationality requirement and use this to discuss the aggregation of experts’ opinions based on probability and (state-independent) utility, showing some inherent limitation of this framework, with implications for statistics. The connection between our results and earlier work in the literature is also discussed.

Journal of Mathematical Psychology, 104, 102582, 2021

## ALEXANDER ERREYGERS, ENRIQUE MIRANDA. A GRAPHICAL STUDY OF COMPARATIVE PROBABILITIES.

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

Synthese, 198(Supp.27), 6529-6570, 2021

## MARCO ZAFFALON, ENRIQUE MIRANDA. DESIRABILITY FOUNDATIONS OF ROBUST RATIONAL DECISION MAKING

Recent work has formally linked the traditional axiomatisation of incomplete preferences à la Anscombe-Aumann with the theory of desirability developed in the context of imprecise probability, by showing in particular that they are the very same theory. The equivalence has been established under the constraint that the set of possible prizes is finite. In this paper, we relax such a constraint, thus de facto creating one of the most general theories of rationality and decision making available today. We provide the theory with a sound interpretation and with basic notions, and results, for the separation of beliefs and values, and for the case of complete preferences. Moreover, we discuss the role of conglomerability for the presented theory, arguing that it should be a rationality requirement under very broad conditions.

International Journal of Approximate Reasoning, 145C, 91-120, 2022

## SÉBASTIEN DESTERCKE, IGNACIO MONTES, ENRIQUE MIRANDA. PROCESSING DISTORTION MODELS: A COMPARATIVE STUDY.

When dealing with sets of probabilities, distortion or neighbourhood models are convenient practical tools, as they rely on very little parameters. In this paper, we study their behaviour when such models are combined and processed through some reasoning tools. 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.

Safety Science, 157, 105915, 2023

## PABLO RAMSÉS ALONSO-MARTÍN, IGNACIO MONTES, ENRIQUE MIRANDA. DISTORTION MODELS FOR ESTIMATING HUMAN ERROR PROBABILITIES

Human Reliability Analysis aims at identifying, quantifying and proposing solutions to human factors causing hazardous consequences. Quantifying the influence of the human factors gives rise to human error probabilities, whose estimation is a cumbersome problem. Since these human factors are usually related to other organisational or technological factors, it has been proposed to apply probabilistic graphical models, such as Bayesian or credal networks. However, these can be problematic when conditional probabilities on missing data are involved. While the solutions proposed so far combine frequentist and subjective approaches and are in general not robust to small modifications in the dataset, in this paper we propose an alternative based on distortion models, which are a type of imprecise probabilities. We perform a comparative analysis, showing that our proposal is consistent with the previous studies while giving rise to robust estimations.

Annals of Operations Research, 321, 409-449, 2023.

## ENRIQUE MIRANDA, IGNACIO MONTES. CENTROIDS OF THE CORE OF EXACT CAPACITIES: A COMPARATIVE STUDY

Capacities are a common tool in decision making. Each capacity determines a core, which is a polytope formed by additive measures. The problem of eliciting a single probability from the core is interesting in a number of fields: in coalitional game theory for selecting a fair way

of splitting the wealth between the players, in the transferable belief model from evidence theory or for transforming a second order into a first order model. In this paper, we study this problem when the goal is to determine the centroid of the core of a capacity, and we compare

four approaches: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the limit of a procedure of uniform contraction. We show that these four centroids do not coincide in general, we give some sufficient conditions for their equality, and we analyse their axiomatic properties.We also discuss how to define a notion of centrality measure indicating the degree of centrality of an additive measure in the core. Finally, we also analyse these four centroids in the more general context of imprecise probabilities.

International Journal of Approximate Reasoning, 154C, 176-199, 2023.

## ENRIQUE MIRANDA, MARCO ZAFFALON. NONLINEAR DESIRABILITY THEORY.

Desirability can be understood as an extension of Anscombe and Aumann's Bayesian decision theory to sets of expected utilities. At the core of desirability lies an assumption of linearity of the scale in which rewards are measured. It is a traditional assumption used to derive the expected utility model, which clashes with a general representation of rational decision making, though. Allais has, in particular, pointed this out in 1953 with his famous paradox. We note that the utility scale plays the role of a closure operator when we regard desirability as a logical theory. This observation enables us to extend desirability to the nonlinear case by letting the utility scale be represented via a general closure operator. The new they directly expresses rewards in actual nonlinear currency (money), much in Savage's spirit, while arguably weakening the founding assumptions to a minimum. We characterise the main properties of the new theory both from the perspective of sets of gambles and of their lower and upper prices (previsions). We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.

Annals of Operations Research, in press.

# ENRIQUE MIRANDA, IGNACIO MONTES, ANDRÉS PRESA. INNER APPROXIMATIONS OF COHERENT LOWER PROBABILITIES AND THEIR APPLICATION TO DECISION MAKING PROBLEMS.

We consider a decision making problem under imprecision, where the probabilistic information is given in terms of a set of probability measures, and where finding the optimal alternative(s) may be difficult. To ease the computation, we propose to transform the initial model into another one that (i) belongs to some subclass with better mathematical properties, such as supermodularity or complete monotonicity; (ii) is at least as informative as the original model, while being as close as possible to it. We show that the problem can be approached in terms of linear or quadratic programming and that it can be connected with the one of determining the incenter of a credal set. Finally, we compare the solutions of a decision making problem with the initial and the transformed models and illustrate how our approach can be applied in a decision making problem under severe uncertainty.

# JOURNAL PUBLICATIONS

The available full papers constitute earlier versions of the published ones

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol 10 (1), pp. 1--15, 2002

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RELATIONSHIPS BETWEEN POSSIBILITY MEASURES AND NESTED RANDOM SETS

Different authors have observed some relationships between consonant random sets and possibility measures, specially for finite universes. In this paper, we go deeply into this matter and propose several possible definitions for the concept of consonant random set. Three of these conditions are equivalent for finite universes. In that case, the random set considered is associated to a possibility measure if and only if any of them is satisfied. However, in a general context, none of the six definitions here proposed is sufficient for a random set to induce a possibility measure. Moreover, only one of them seems to be necessary.

International Journal of Approximate Reasoning, 32(1), pp. 23--42, 2003.

## ENRIQUE MIRANDA, GERT DE COOMAN. EPISTEMIC INDEPENDENCE IN NUMERICAL POSSIBILITY THEORY

Numerical possibility measures can be interpreted as systems of upper betting rates for events. As such, they have a special part in the unifying behavioural theory of imprecise probabilities, proposed by Walley. On this interpretation, they should arguably satisfy certain rationality, or consistency, requirements, such as avoiding sure loss and coherence. Using a version of Walley's notion of epistemic independence suitable for possibility measures, we study in detail what these rationality requirements tell us about the construction of independent product possibility measures from given marginals, and we obtain necessary and sufficient conditions for a product to satisfy these criteria. In particular, we show that the well-known minimum and product rules for forming independent joint distributions from marginal ones, are only coherent when at least one of these distributions assume just the values zero and one.

International Journal of Approximate Reasoning, Vol 33(1), pp. 95--115, 2003.

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, EXTREME POINTS OF CREDAL SETS GENERATED BY 2-ALTERNATING CAPACITIES

The characterization of the extreme points constitutes a crucial issue in the investigation of convex sets of probabilities, not only from a purely theoretical point of view, but also as a tool in the management of imprecise information. In this respect, different authors have found an interesting relation between the extreme points of the class of probability measures dominated by a second order alternating Choquet capacity and the permutations of the elements in the referential. However, they have all restricted their work to the case of a finite referential space. In an infinite setting, some technical complications arise and they have to be carefully treated. In this paper, we extend the mentioned result to the more general case of separable metric spaces. Furthermore, we derive some interesting topological properties about the convex sets of probabilities here investigated. Finally, a closer look to the case of possibility measures is given: for them, we prove that the number of extreme points can be reduced even in the finite case.

Information Sciences, Vol 168(1--4), pp. 51--75, 2004

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, A RANDOM SET CHARACTERISATION OF POSSIBILITY MEASURES

Several authors have pointed out the relationship between consonant random sets and possibility measures. However, this relationship has only been proven for the finite case, where the inverse Möbius of the upper probability induced by the random set simplifies the computations to a great extent. In this paper, we study the connection between both concepts for arbitrary referential spaces. We complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases.

Journal of Statistical Planning and Inference, Vol 133(1), pp. 173-197, 2005.

## ENRIQUE MIRANDA, GERT DE COOMAN AND INÉS COUSO, LOWER PREVISIONS INDUCED BY MULTI-VALUED MAPPINGS

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 their explanatory and unifying power.

Journal of Mathematical Analysis and Applications, Vol 307(1), pp. 32-47, 2005

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RANDOM SETS AS IMPRECISE RANDOM VARIABLES

Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the probability 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 probabilities induced by the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.

Fuzzy Sets and Systems, Vol 154(3), pp. 386-412, 2005.

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL, RANDOM INTERVALS AS A MODEL FOR IMPRECISE INFORMATION

Random intervals constitute one of the classes of random sets with a greater number of applications. In this paper, we regard them as the imprecise observation of a random variable, and study how to model the information about the probability distribution of this one. Two possible models are the probability distributions of the measurable selections and those bounded by the upper probability. We prove that, under some hypotheses, the closures of these two sets in the topology of the weak convergence coincide, improving results from the literature. Moreover, we provide examples showing that the two models are not equivalent in general, and give sufficient conditions for the equality between them. Finally, we comment on the relationship between random intervals and fuzzy numbers.

Journal of Intelligent and Fuzzy Systems, Vol 16(4) (Special Issue dedicated to the 60th birthday of Etienne E. Kerre), pp. 253-263, 2005.

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA. N-MONOTONE LOWER PREVISIONS

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.

Journal of Theoretical Probability, Vol 20(3), 663-693, 2007

## ENRIQUE MIRANDA, GERT DE COOMAN AND ERIK QUAEGHEBEUR, THE HAUSDORFF MOMENT PROBLEM UNDER FINITE ADDITIVITY

We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.

International Journal of Approximate Reasoning, Vol 46(1), 188-225, 2007

## ENRIQUE MIRANDA AND GERT DE COOMAN, MARGINAL EXTENSION IN THE THEORY OF COHERENT LOWER PREVISIONS

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

Journal of Mathematical Analysis and Applications, 340 (2), 982-999, 2008

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA, A UNIFYING APPROACH TO INTEGRATION FOR BOUNDED POSITIVE CHARGES

This paper deals with n-monotone functionals, which constitute a generalisation of n-monotone set functions. Using the notion of exactness of a functional, we introduce a new notion of lower and upper integral which subsumes as particular cases most of the approaches to integration in the literature. As a consequence, we can characterise which types of integrals can be used to calculate the natural extension (the lower envelope of all linear extensions) of a positive bounded charge.

International Journal of Approximate Reasoning, 48(1), 132-155, 2008

## ENRIQUE MIRANDA, GERT DE COOMAN AND ERIK QUAEGHEBEUR, FINITELY ADDITIVE EXTENSIONS OF DISTRIBUTION FUNCTIONS AND MOMENT SEQUENCES: THE COHERENT LOWER PREVISION APPROACH

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions.

International Journal of Approximate Reasoning, 48(2), 628-658, 2008

## ENRIQUE MIRANDA, A SURVEY OF THE THEORY OF COHERENT LOWER PREVISIONS.

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.

Journal of Statistical Planning and Inference, 138(8), 2409-2432, 2008

## GERT DE COOMAN AND ENRIQUE MIRANDA, WEAK AND STRONG LAWS OF LARGE NUMBERS FOR COHERENT LOWER PREVISIONS

We prove weak and strong laws of large numbers for coherent lower previsions, where the lower prevision of a random variable is given a behavioural interpretation as a subject's supremum acceptable price for buying it. Our laws are a consequence of the rationality criterion of coherence, and they can be proven under assumptions that are surprisingly weak when compared to the standard formulation of the laws in more classical approaches to probability theory.

Journal of Mathematical Analysis and Applications, 347(1), 143--156, 2008

## GERT DE COOMAN, MATTHIAS TROFFAES AND ENRIQUE MIRANDA. N-MONOTONE EXACT FUNCTIONALS

We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals.

Artificial Intelligence, 173(1), 104--144, 2009

## ENRIQUE MIRANDA AND MARCO ZAFFALON. COHERENCE GRAPHS

We study the consistency of a number of probability distributions, which are allowed to be imprecise. To make the treatment as general as possible, we represent those probabilistic assessments as a collection of conditional lower previsions. The problem then becomes proving Walley's (strong) coherence of the assessments. In order to maintain generality in the analysis, we assume to be given nearly no information about the numbers that make up the lower previsions in the collection. Under this condition, we investigate the extent to which the above global task can be decomposed into simpler and more local ones. This is done by introducing a graphical representation of the conditional lower previsions that we call the coherence graph: we show that the coherence graph allows one to isolate some subsets of the collection whose coherence is sufficient for the coherence of all the assessments; and we provide a polynomial-time algorithm that finds the subsets efficiently. We show some of the implications of our results by focusing on three models and problems: Bayesian and credal networks, of which we prove coherence; the compatibility problem, for which we provide an optimal graphical decomposition; probabilistic satisfiability, of which we show that some intractable instances can instead be solved efficiently by exploiting coherence graphs

Journal of Statistical Planning and Inference, 139(2), 256-276, 2009

## GERT DE COOMAN AND ENRIQUE MIRANDA, FORWARD IRRELEVANCE

We investigate how to combine marginal assessments about the values that random variables assume separately into a model for the values that they assume jointly, when (i) these marginal assessments are modelled by means of coherent lower previsions, and (ii) we have the additional assumption that the random variables are forward epistemically irrelevant to each other. We consider and provide arguments for two possible combinations, namely the forward irrelevant natural extension and the forward irrelevant product, and we study the relationships between them. 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

International Journal of Approximate Reasoning, 50(2), special issue on the Imprecise Dirichlet Model, 204-216, 2009

## GERT DE COOMAN, ENRIQUE MIRANDA AND ERIK QUAEGHEBEUR, REPRESENTATION INSENSITIVITY IN IMMEDIATE PREDICTION

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. We establish that such representation insensitive predictive models have very interesting properties, and show that among such models, the ones produced by the Imprecise Dirichlet (Multinomial) Model are quite special in a number of ways. In the Conclusion, we discuss the open question as to how unique the Imprecise Dirichlet (Multinomial) Model predictive lower previsions are in being representation insensitive.

Fuzzy Sets and Systems, 160(9), 1286-1307, 2009

## ENRIQUE MIRANDA, UPDATING COHERENT PREVISIONS ON FINITE SPACES

We compare the different notions of conditional 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, and we discuss the extensions of our results to infinite spaces.

Journal of Artificial Intelligence Research, 34, 757-821, 2009

## MARCO ZAFFALON AND ENRIQUE MIRANDA. CONSERVATIVE INFERENCE RULE FOR UNCERTAIN REASONING UNDER INCOMPLETENESS

In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process' behaviour to be partly unknown. Then we use Walley's theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classi cation, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences.

Bernoulli, 15(3), 721-735, 2009

## GERT DE COOMAN, ERIK QUAEGHEBEUR AND ENRIQUE MIRANDA, EXCHANGEABLE LOWER PREVISIONS

We extend de Finetti's notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We derive representation theorems in both the finite and the countable case, in terms of sampling without and with replacement, respectively.

Information Sciences, 180(8), 1407-1417, 2010

## ENRIQUE MIRANDA, INÉS COUSO AND PEDRO GIL. APPROXIMATION OF UPPER AND LOWER PROBABILITIES BY MEASURABLE SELECTIONS

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. In doing this, we generalise a number of results from the literature. Finally, we study the particular case of consonant random sets and we also derive a relationship between Aumann and Choquet integrals.

Journal of Statistical Planning and Inference, 140(7), 1805-1833, 2010

## ENRIQUE MIRANDA AND MARCO ZAFFALON, CONDITIONAL MODELS: COHERENCE AND INFERENCE THROUGH SEQUENCES OF JOINT MASS FUNCTIONS

We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus on two problems: (coherence) giving conditions to guarantee that a conditional model is self-consistent; (inference) delivering methods to derive new probabilistic statements from a self-consistent conditional model. We address these problems in the case where the probabilistic statements can be specified imprecisely through sets of probabilities, while restricting the attention to finite spaces of possibilities. Using Walley's theory of coherent lower previsions, we fully characterise the question of coherence, and specialise it for the case of precisely specified probabilities, which is the most common case addressed in the literature. This shows that coherent conditional models are equivalent to sequences of (possibly sets of) unconditional mass functions. In turn, this implies that the inferences from a conditional model are the limits of the conditional inferences obtained by applying Bayes' rule, when possible, to the elements of the sequence. In doing so, we unveil the tight connection between conditional models and zero-probability events. Such a connection appears to have been overlooked by most previous works on the subject, thus preventing so far to give a full account of coherence and inference for conditional models.

Annals of Mathematics and Artificial Intelligence, 60(3-4), 251-309, 2010

## ENRIQUE MIRANDA AND MARCO ZAFFALON. NOTES ON DESIRABILITY AND CONDITIONAL LOWER PREVISIONS

We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley's celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional lower previsions is derived from a set of desirable gambles, and its converse. In either case, we relate the properties of the derived model with those of the originating one. Our results constitute basic tools to move from one formalism to the other, and thus to take advantage of work done in the two fronts. Sets of desirable gambles are at the same time very powerful and intuitive models of uncertainty. Given the central role of uncertainty in artificial intelligence, this work marks a key passage towards the wider accessibility of those modelling capabilities in artificial intelligence.

IEEE Transactions on Automatic Control, 56(7), 1567--1581, 2011

## ALESSIO BENAVOLI, MARCO ZAFFALON AND ENRIQUE MIRANDA, A NEW ROBUST APPROACH TO FILTERING BASED ON COHERENT LOWER PREVISIONS

The classical filtering problem is re-examined to take into account imprecision in the knowledge about the probabilistic relationships involved. To achieve that, we consider closed convex 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.

Artificial Intelligence, 175(12-13), 1911--1950, 2011

## GERT DE COOMAN, ENRIQUE MIRANDA AND MARCO ZAFFALON. INDEPENDENT NATURAL EXTENSION

There is no unique extension of the standard notion of probabilistic independence to the case where probabilities are indeterminate or imprecisely specified. Epistemic independence is an extension that formalises the intuitive idea of mutual irrelevance between different sources of information. It has a wide scope and great appeal, especially for a field like Artificial Intelligence, where such an idea or interpretation of independence has already been employed quite often in precise-probabilistic contexts. Despite of this, epistemic independence has received little attention so far. This paper develops its foundations for variables assuming values in finite spaces. We define (epistemically) independent products of marginals (or possibly conditionals) and show that there always is a unique least-committal such independent product, which we call the independent natural extension.We supply an explicit formula for it, and study some of its properties: associativity, marginalisation, external additivity, which are basic tools to work with the independent natural extension. Additionally, we consider a number of ways in which the standard factorisation formula for independence can be generalised to an imprecise-probabilistic context. We show, under some mild conditions, that when the focus is on least-committal models, using the independent natural extension is equivalent to imposing a so-called strong factorisation property. This is an important outcome for applications as it gives a simple tool to make sure that inferences are consistent with epistemic independence judgments. We discuss the potential of our results for applications in Artificial Intelligence by recalling recent work by some of us, where the independent natural extension was applied to graphical models. It has allowed, for the first time, the development of an exact linear-time algorithm for the imprecise-probability updating of credal trees.

International Journal of Approximate Reasoning, 53(8), 1200--1227, 2012

## ENRIQUE MIRANDA, MARCO ZAFFALON AND GERT DE COOMAN. CONGLOMERABLE NATURAL EXTENSION

At the foundations of probability theory lies a question that has been open since de Finetti framed it in 1930: whether or not an uncertainty model should be required to be conglomerable. Conglomerability is related to accepting infinitely many conditional bets. Walley is one of the authors who have argued in favor of conglomerability, while de Finetti rejected the idea. In this paper we study the extension of the conglomerability condition to two types of uncertainty models that are more general than the ones envisaged by de Finetti: sets of desirable gambles and coherent lower previsions. We focus in particular on the weakest (i.e., the least-committal) of those extensions, which we call the conglomerable natural extension. The weakest extension that does not take conglomerability into account is simply called the natural extension. We show that taking the natural extension of assessments after imposing conglomerability---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 of imposing conglomerability and taking the natural extension produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether this sequence converges to it. We give sufficient conditions for this to happen in some special cases, and study the differences between working with coherent sets of desirable gambles and coherent lower previsions. Our results indicate that it is necessary to rethink the foundations of Walley's theory of coherent lower previsions for infinite partitions of conditioning events

Journal of Artificial Intelligence Research, 45, 601--640, 2012

## GERT DE COOMAN AND ENRIQUE MIRANDA. IRRELEVANT AND INDEPENDENT NATURAL EXTENSION FOR SETS OF DESIRABLE GAMBLES

The results in this paper add useful tools to the theory of sets of desirable gambles, a growing toolbox for reasoning with partial probability assessments. 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 coherent sets of desirable 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. Having such a set of tools that are easily implemented in computer programs is clearly beneficial to fields, like AI, with a clear interest in coherent reasoning under uncertainty using general and robust uncertainty models that require no full specification

Information Sciences, 224, 88--108, 2013

## MATTHIAS TROFFAES, ENRIQUE MIRANDA AND SÉBASTIEN DESTERCKE. ON THE CONNECTION BETWEEN PROBABILITY BOXES AND POSSIBILITY MEASURES

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

Artificial Intelligence, 198, 1--51, 2013

## MARCO ZAFFALON AND ENRIQUE MIRANDA. PROBABILITY AND TIME

Probabilistic reasoning is often attributed a temporal meaning, in which conditioning is regarded as a normative rule to compute future beliefs out of current beliefs and observations. However, the well-established ‘updating interpretation’of conditioning is not concerned with beliefs that evolve in time, and in particular with future beliefs. On the other hand, a temporal justification of conditioning was proposed already by De Moivre and Bayes, by requiring that current and future beliefs be consistent. We reconsider the latter proposal while dealing with a generalised version of the problem, using a behavioural theory of imprecise probability in the form of coherent lower previsions as well as of coherent sets of desirable gambles, and letting the possibility space be finite or infinite. We obtain that using conditioning is normative, in the imprecise case, only if one establishes future behavioural commitments at the same time of current beliefs. In this case it is also normative that present beliefs be conglomerable, which is a result that touches on a long-term controversy at the foundations of probability. In the remaining case, where one commits to some future behaviour after establishing present beliefs, we characterise the several possibilities to define consistent future assessments; this shows in particular that temporal consistency does not preclude changes of mind. And yet, our analysis does not support that rationality requires consistency in general, even though pursuing consistency makes sense and is useful, at least as a way to guide and evaluate the assessment process. These considerations narrow down in the special case of precise probability, because this formalism cannot distinguish the two different situations illustrated above: it turns out that the only consistent rule is conditioning and moreover that it is not rational to be willing to stick to precise probability while using a rule different from conditioning to compute future beliefs; rationality requires in addition the disintegrability of the present-time probability.

International Journal of Approximate Reasoning, 54(9), 1322--1350, 2013

## ENRIQUE MIRANDA AND MARCO ZAFFALON. CONGLOMERABLE COHERENCE

We contrast Williams' and Walley's theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them the most: Williams' theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walley's notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considered also some previous results in the literature, we deduce that Williams' theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walley's theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.

Computational Statistics and Data Analysis, 71(C), 867-885, 2014

## IGNACIO MONTES, ENRIQUE MIRANDA AND SUSANA MONTES. STOCHASTIC DOMINANCE WITH IMPRECISE INFORMATION

Stochastic dominance, which is based on the comparison of distribution functions, is one of the most popular preference measures. However, its use is limited to the case where the goal is to compare pairs of distribution functions, whereas in many cases it is interesting to compare sets of distribution functions: this may be the case for instance when the available information does not allow to fully elicitate the probability distributions of the random variables. To deal with these situations, a number of generalisations of the notion of stochastic dominance are proposed; their connection with an equivalent p-box representation of the sets of distribution functions is studied; a number of particular cases, such as sets of distributions associated to possibility measures, are investigated; and an application to the comparison of the Lorenz curves of countries within the same region is presented

Journal of Mathematical Analysis and Applications, 410(1), 101--116, 2014

## GERT DE COOMAN AND ENRIQUE MIRANDA. LOWER PREVISIONS INDUCED BY FILTER MAPS

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.

European Journal of Operational Research, 234(1), 209--220, 2014

## IGNACIO MONTES, ENRIQUE MIRANDA AND SUSANA MONTES. DECISION MAKING WITH IMPRECISE UTILITIES AND BELIEFS BY MEANS OF STATISTICAL PREFERENCE AND STOCHASTIC DOMINANCE

A problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered. A number of choice rules are proposed and their main properties are investigated, focusing particularly on the generalizations of stochastic dominance and statistical preference. The particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered.

International Journal of Approximate Reasoning, 56(A), 1--27, 2015

## ENRIQUE MIRANDA, MARCO ZAFFALON. ON THE PROBLEM OF COMPUTING THE CONGLOMERABLE NATURAL EXTENSION

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension, provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension. In this paper we consider probabilistic assessments in the form of a coherent lower prevision P, which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F ≥ P that is conglomerable, in case it exists. Past work has shown that F can be approximated by an increasing sequence (F n)n of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many 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

International Journal of Approximate Reasoning, 56(B), 159-177, 2015

## ENRIQUE MIRANDA, IGNACIO MONTES. COHERENT UPDATING OF NON-ADDITIVE MEASURES

The conditions under which a 2-monotone lower prevision can 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, which are shown to correspond to the particular case of vacuous lower previsions.

Fuzzy Sets and Systems, 278C, 48-66, 2015

## IGNACIO MONTES, ENRIQUE MIRANDA, RENATO PELESSONI, PAOLO VICIG. SKLAR'S THEOREM IN AN IMPRECISE SETTING

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 single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

Information Sciences, 298, 373-389, 2015

## ENRIQUE MIRANDA, MATTHIAS TROFFAES, SÉBASTIEN DESTERCKE. A GEOMETRIC AND GAME-THEORETIC STUDY OF THE CONJUNCTION OF POSSIBILITY MEASURES

Journal of Mathematical Psychology, 64/65, 44-57, 2015

## ENRIQUE MIRANDA, SÉBASTIEN DESTERCKE. EXTREME POINTS OF THE CREDAL SETS GENERATED BY COMPARATIVE PROBABILITIES

When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixture 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.

Journal of Mathematical Analysis and Applications, 425/1, 460-488, 2015

## ENRIQUE MIRANDA, MARCO ZAFFALON. INDEPENDENT PRODUCTS IN INFINITE SPACES

Probabilistic independence, intended as the mutual irrelevance of given variables, can be solidly founded on a notion of self-consistency of an uncertainty model, in particular when probabilities go imprecise. There is nothing in this approach that prevents it from being adopted in very general setups, and yet it has mostly been detailed for variables taking finitely many values. In this mathematical study, we complement previous research by exploring the extent to which such an approach can be generalised. We focus in particular on the independent products of two variables. We characterise the main notions, including some of factorisation and productivity, in the general case where both spaces can be infinite and show that, however, there are situations---even in the case of precise probability---where no independent product exists. This is not the case as soon as at least one space is finite. We study in depth this case at the frontiers of well-behaviour detailing the relations among the most important notions; we show for instance that being an independent product is equivalent to a certain productivity condition. Then we step back to the general case: we give conditions for the existence of independent products and study ways to get around its inherent limitations.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 24/2, 229-263, 2016

## RENATO PELESSONI, PAOLO VICIG, IGNACIO MONTES, ENRIQUE MIRANDA. BIVARIATE P-BOXES

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) equivalent to 2-coherence. We therefore focus on the relevant subclass of coherent p-boxes, corresponding to coherent lower probabilities on special domains. Several properties of coherent p-boxes are investigated and compared with those of (one-dimensional) p-boxes or of bivariate distribution functions.

International Journal of Approximate Reasoning, 78/C, 125-137, 2016

## ENRIQUE MIRANDA, MARCO ZAFFALON. CONFORMITY AND INDEPENDENCE WITH COHERENT LOWER PREVISIONS

We define the conformity of marginal and conditional models with a joint model within Walley's theory of coherent lower previsions. Loosely speaking, conformity means that the joint can reproduce the marginal and conditional models we started from. By studying conformity with and without additional assumptions of epistemic irrelevance and independence, we establish connections with a number of prominent models in Walley's theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.

Fuzzy Optimization and Decision Making, 16/4, 297-327, 2017

## IGNACIO MONTES, ENRIQUE MIRANDA, SUSANA MONTES. IMPRECISE STOCHASTIC ORDERS AND FUZZY RANKINGS

We extend the notion of stochastic order to the pairwise comparison of fuzzy random variables. We consider expected utility, stochastic dominance and statistical preference, which are related to the comparisons of the expectations, distribution functions and medians of the underlying variables, and discuss how to generalize these notions to the fuzzy case, when an epistemic interpretation is given to the fuzzy random variables. In passing, we investigate to which extent the earlier extensions of stochastic dominance and expected utility to the comparison of sets of random variables can be useful as fuzzy rankings.

Journal of Statistical Theory and Practice, 11/4, 634-669, 2017

## ENRIQUE MIRANDA, MARCO ZAFFALON. FULL CONGLOMERABILITY

We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley in his 1991 book. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, super-additivity and marginal extension. Moreover, we discuss the special case where a model is conglomerable with respect to a subset of all the partitions, and try to sort out the different notions of conglomerability present in the literature. We conclude that countable additivity, which is routinely used to impose full conglomerability in the precise case, appears to be the most well-behaved way to do so in the imprecise case as well by taking envelopes of countably additive probabilities. Moreover, we characterise these envelopes by means of a number of necessary and sufficient conditions.We do a thorough mathematical study of the notion of full conglomerability, that is, conglomerability with respect to all the partitions of an infinite possibility space, in the sense considered by Peter Walley in his 1991 book. We consider both the cases of precise and imprecise probability (sets of probabilities). We establish relations between conglomerability and countable additivity, continuity, super-additivity and marginal extension. Moreover, we discuss the special case where a model is conglomerable with respect to a subset of all the partitions, and try to sort out the different notions of conglomerability present in the literature. We conclude that countable additivity, which is routinely used to impose full conglomerability in the precise case, appears to be the most well-behaved way to do so in the imprecise case as well by taking envelopes of countably additive probabilities. Moreover, we characterise these envelopes by means of a number of necessary and sufficient conditions.

International Journal of General Systems, 46/4, 354-385, 2017

## IGNACIO MONTES, ENRIQUE MIRANDA. BIVARIATE P-BOXES AND MAXITIVE FUNCTIONS

We give necessary and sufficient conditions for a maxitive function to be the upper probability of a bivariate p-box, in terms of its associated possibility distribution and its focal sets. This allows us to derive conditions in terms of the lower and upper distribution functions of the bivariate p-box. In particular, we prove that only bivariate p-boxes with a non-informative lower or upper distribution function may induce a maxitive function. In addition, we also investigate the extension of Sklar’s theorem to this context.

Fuzzy Sets and Systems, 341C, 1-36, 2018

## ARTHUR VAN CAMP, GERT DE COOMAN, ENRIQUE MIRANDA, ERIK QUAEGHEBEUR. COHERENT CHOICE FUNCTIONS, DESIRABILITY AND INDIFFERENCE

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 definition of indifference in terms of choice functions, which we discuss in some detail.

International Journal of Approximate Reasoning, 88/C, 387-400, 2017

## PATRIZIA BERTI, ENRIQUE MIRANDA, PIETRO RIGO. BASIC IDEAS UNDERLYING CONGLOMERABILITY AND DISINTEGRABILITY

The basic mathematical theory underlying the notions of conglomerability and disintegrability is reviewed. Both the precise and the imprecise cases are concerned.

Journal of Artificial Intelligence Research, 60, 1057-1126, 2017

## MARCO ZAFFALON, ENRIQUE MIRANDA. AXIOMATISING INCOMPLETE PREFERENCES THROUGH SETS OF DESIRABLE GAMBLES

We establish the equivalence of two very general theories: the first is the decision-theoretic formalisation of incomplete preferences based on the mixture independence axiom; the second is the theory of coherent sets of desirable gambles (bounded variables) developed in the context of imprecise probability and extended here to vector-valued gambles. Such an equivalence allows us to analyse the theory of incomplete preferences from the point of view of desirability. Among other things, this leads us to uncover an unexpected and clarifying relation: that the notion of state independence—the traditional assumption that we can have separate models for beliefs (probabilities) and values (utilities)—coincides with that of strong independence in imprecise probability; this connection leads us also to propose much weaker, and arguably more realistic, notions of state independence. Then we simplify the treatment of complete beliefs and values by putting them on a more equal footing. We study the role of the Archimedean condition—which allows us to actually talk of expected utility—, identify some weaknesses and propose alternatives that solve these. More generally speaking, we show that desirability is a valuable alternative foundation to preferences for decision theory that streamlines and unifies a number of concepts while preserving great generality. In addition, the mentioned equivalence shows for the first time how to extend the theory of desirability to imprecise non-linear utility, thus enabling us to formulate one of the most powerful self-consistent theories of reasoning and decision-making available today.

International Journal of Approximate Reasoning, 92, 97-119, 2018

## ARTHUR VAN CAMP, GERT DE COOMAN, ENRIQUE MIRANDA. LEXICOGRAPHIC CHOICE FUNCTIONS

We investigate a generalisation of the coherent choice functions considered by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no Archimedeanity condition. We define our choice functions on vector spaces of options, which allows us to incorporate as special cases both Seidenfeld et al.’s (2010) choice functions on horse lotteries and also pairwise choice—which is equivalent to sets of desirable gambles (Quaeghebeur, 2014)—, and to investigate their connections. We show that choice functions based on sets of desirable options (gambles) satisfy Seidenfeld’s convexity axiom only for very particular types of sets of desirable options, which are exactly those that are representable by lexicographic probability systems that have no non-trivial Savage-null events. We call them lexicographic choice functions. Finally, we prove that these choice functions can be used to determine the most conservative convex choice function associated with a given binary relation.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26/6, 1017-1051, 2018

## IGNACIO MONTES, ENRIQUE MIRANDA. EXTREME POINTS OF THE CORE OF POSSIBILITY MEASURES AND P-BOXES

Under an epistemic interpretation, an upper probability can be regarded as equivalent to the set of probability measures it dominates, sometimes referred to as its core. In this paper, we study the properties of the number of extreme points of the core of a possibility measure, and investigate in detail those associated with (uni- and bi-)variate p-boxes, that model the imprecise information about a cumulative distribution function.

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26/6, 917-947, 2018

## ENRIQUE MIRANDA, IGNACIO MONTES. SHAPLEY AND BANZHAF VALUES AS PROBABILITY TRANSFORMATIONS

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations. 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 consider its normalized version. We study a number of particular models of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distortion models. For them, we provide some alternative expressions of the Shapley and Banzhaf values and study under which conditions they belong to the core of the lower probability.

International Journal of Approximate Reasoning, 101, 181-205, 2018

## IGNACIO MONTES, ENRIQUE MIRANDA, PAOLO VICIG. 2-MONOTONE OUTER APPROXIMATIONS OF COHERENT LOWER PROBABILITIES

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: quadratic programming, extensions of the total variation distance, and the Weber set from game theory. Finally, we show that our work applies to the more general problem of approximating coherent lower previsions.

Information Sciences, 481, 550-573, 2019

## IGNACIO MONTES, ENRIQUE MIRANDA, SÉBASTIEN DESTERCKE. PARI-MUTUEL PROBABILITIES AS AN UNCERTAINTY MODEL

The pari-mutuel model is a betting scheme that has its origins in horse racing, and that has been applied in a number of contexts, mostly economics. In this paper, we consider the set of probability measures compatible with a pari-mutuel model, characterize its extreme points, and investigate the properties of the associated lower and upper probabilities. We show that the pari-mutuel model can be embedded within the theory of probability intervals, and prove necessary and sufficient conditions for it to be a belief function or a minitive measure. In addition, we also investigate the combination of different pari-mutuel models and their definition on product spaces.

International Journal of Approximate Reasoning, 110, 1-30, 2019

## IGNACIO MONTES, ENRIQUE MIRANDA, PAOLO VICIG. OUTER APPROXIMATING COHERENT LOWER PROBABILITIES WITH BELIEF FUNCTIONS

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 the same time as close as possible to the initial coherent lower probability while not including additional information. We show that this problem can be tackled by means of linear programming, and investigate the features of the set of optimal solutions. Moreover, we emphasize the differences with the outer approximations by 2-monotone lower probabilities. We also study the problem for two particular cases of belief functions that are computationally easier to handle: necessity measures and probability boxes.

Artificial Intelligence, 283C, 2020

## ENRIQUE MIRANDA, MARCO ZAFFALON. COMPATIBILITY, DESIRABILITY AND THE RUNNING INTERSECTION PROPERTY

Compatibility is the problem of checking whether some given probabilistic assessments have a common joint probabilistic model. When the assessments are unconditional, the problem is well established in the literature and finds a solution through the running intersection property (RIP). This is not the case of conditional assessments. In this paper, we study the compatibility problem in a very general setting: any possibility space, unrestricted domains, imprecise (and possibly degenerate) probabilities. We extend the unconditional case to our setting, thus generalising most of previous results in the literature. The conditional case turns out to be fundamentally different from the unconditional one. For such a case, we prove that the problem can still be solved in general by (RIP) but in a more involved way: by constructing a junction tree and propagating information over it. Still, (RIP) does not allow us to optimally take advantage of sparsity: in fact, conditional compatibility can be simplified further by joining junction trees with coherence graphs.

International Journal of General Systems, 49/6, 602-635, 2020

## IGNACIO MONTES, ENRIQUE MIRANDA, AND SÉBASTIEN DESTERCKE. UNIFYING NEIGHBOURHOOD AND DISTORTION MODELS. PART I: NEW RESULTS ON OLD MODELS

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, the linear vacuous mixtures or the constant odds ratio model. This paper is the first part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods defined over specific metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This first part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our findings.

The version below includes a small correction in the convexity axiom Ax.4, that does not affect the validity of the results.

International Journal of General Systems, 49/6, 636-674, 2020

## IGNACIO MONTES, ENRIQUE MIRANDA AND SÉBASTIEN DESTERCKE. UNIFYING NEIGHBOURHOOD AND DISTORTION MODELS. PART II: NEW MODELS AND SYNTHESIS

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. In the first part of this study, we introduced a general, unified view encompassing such neighbourhoods, and revisited some well-known models (pari mutuel, linear vacuous, constant odds-ratio). In this second part, we study models that have received little to no attention, but are induced by classical distances between probabilities, such as the total variation, the Kolmogorov and the L 1 distances. We finish by comparing those models in terms of a number of properties: precision, number of extreme points, n-monotonicity, ...thus providing possible guidelines for selecting a neighbourhood rather than another.

International Journal of Approximate Reasoning, 125C, 49-72, 2020

## ARTHUR VAN CAMP, ENRIQUE MIRANDA. MODELLING EPISTEMIC IRRELEVANCE WITH CHOICE FUNCTIONS.

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.

Fuzzy Sets and Systems, 424C, 1-36, 2021

## ENRIQUE MIRANDA, IGNACIO MONTES, PAOLO VICIG. ON THE SELECTION OF AN OPTIMAL OUTER APPROXIMATION OF A COHERENT LOWER PROBABILITY.

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 studied the problem of outer approximating a given coherent lower probability by a more tractable model, such as a 2- or completely monotone lower probability. Unfortunately, such an outer approximation is not unique in general, even if we restrict our attention to those that are undominated by other models from the same family. In this paper, we investigate whether a number of approaches may help in selecting a unique undominated outer approximation. These are based on minimising a distance with respect to the initial model, maximising the specificity, or preserving the same preferences as the original model. We apply them to 2-and completely monotone approximating lower probabilities, and also to the particular cases of possibility measures and p-boxes.

Annals of Mathematics and Artificial Intelligence, 89(10-11), 965-1011, 2021

## ARIANNA CASANOVA, ENRIQUE MIRANDA, MARCO ZAFFALON. JOINT DESIRABILITY FOUNDATIONS OF SOCIAL CHOICE AND OPINION POOLING

We develop joint foundations for the fields of social choice and opinion pooling using coherent sets of desirable gambles, a general uncertainty model that allows to encompass both complete and incomplete preferences. 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 using the same framework to analyse opinion pooling. In particular, we argue that weak Pareto (unanimity) should be given the status of a rationality requirement and use this to discuss the aggregation of experts’ opinions based on probability and (state-independent) utility, showing some inherent limitation of this framework, with implications for statistics. The connection between our results and earlier work in the literature is also discussed.

Journal of Mathematical Psychology, 104, 102582, 2021

## ALEXANDER ERREYGERS, ENRIQUE MIRANDA. A GRAPHICAL STUDY OF COMPARATIVE PROBABILITIES.

We consider a set of comparative probability judgements over a nite 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

Synthese, 198(Supp.27), 6529-6570, 2021

## MARCO ZAFFALON, ENRIQUE MIRANDA. DESIRABILITY FOUNDATIONS OF ROBUST RATIONAL DECISION MAKING

Recent work has formally linked the traditional axiomatisation of incomplete preferences à la Anscombe-Aumann with the theory of desirability developed in the context of imprecise probability, by showing in particular that they are the very same theory. The equivalence has been established under the constraint that the set of possible prizes is finite. In this paper, we relax such a constraint, thus de facto creating one of the most general theories of rationality and decision making available today. We provide the theory with a sound interpretation and with basic notions, and results, for the separation of beliefs and values, and for the case of complete preferences. Moreover, we discuss the role of conglomerability for the presented theory, arguing that it should be a rationality requirement under very broad conditions.

International Journal of Approximate Reasoning, 145C, 91-120, 2022

## SÉBASTIEN DESTERCKE, IGNACIO MONTES, ENRIQUE MIRANDA. PROCESSING DISTORTION MODELS: A COMPARATIVE STUDY.

When dealing with sets of probabilities, distortion or neighbourhood models are convenient practical tools, as they rely on very little parameters. In this paper, we study their behaviour when such models are combined and processed through some reasoning tools. 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.

Safety Science, 157, 105915, 2023

## PABLO RAMSÉS ALONSO-MARTÍN, IGNACIO MONTES, ENRIQUE MIRANDA. DISTORTION MODELS FOR ESTIMATING HUMAN ERROR PROBABILITIES

Human Reliability Analysis aims at identifying, quantifying and proposing solutions to human factors causing hazardous consequences. Quantifying the influence of the human factors gives rise to human error probabilities, whose estimation is a cumbersome problem. Since these human factors are usually related to other organisational or technological factors, it has been proposed to apply probabilistic graphical models, such as Bayesian or credal networks. However, these can be problematic when conditional probabilities on missing data are involved. While the solutions proposed so far combine frequentist and subjective approaches and are in general not robust to small modifications in the dataset, in this paper we propose an alternative based on distortion models, which are a type of imprecise probabilities. We perform a comparative analysis, showing that our proposal is consistent with the previous studies while giving rise to robust estimations.

Annals of Operations Research, 321, 409-449, 2023.

## ENRIQUE MIRANDA, IGNACIO MONTES. CENTROIDS OF THE CORE OF EXACT CAPACITIES: A COMPARATIVE STUDY

Capacities are a common tool in decision making. Each capacity determines a core, which is a polytope formed by additive measures. The problem of eliciting a single probability from the core is interesting in a number of fields: in coalitional game theory for selecting a fair way

of splitting the wealth between the players, in the transferable belief model from evidence theory or for transforming a second order into a first order model. In this paper, we study this problem when the goal is to determine the centroid of the core of a capacity, and we compare

four approaches: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the limit of a procedure of uniform contraction. We show that these four centroids do not coincide in general, we give some sufficient conditions for their equality, and we analyse their axiomatic properties.We also discuss how to define a notion of centrality measure indicating the degree of centrality of an additive measure in the core. Finally, we also analyse these four centroids in the more general context of imprecise probabilities.

International Journal of Approximate Reasoning, 154C, 176-199, 2023.

## ENRIQUE MIRANDA, MARCO ZAFFALON. NONLINEAR DESIRABILITY THEORY.

Desirability can be understood as an extension of Anscombe and Aumann's Bayesian decision theory to sets of expected utilities. At the core of desirability lies an assumption of linearity of the scale in which rewards are measured. It is a traditional assumption used to derive the expected utility model, which clashes with a general representation of rational decision making, though. Allais has, in particular, pointed this out in 1953 with his famous paradox. We note that the utility scale plays the role of a closure operator when we regard desirability as a logical theory. This observation enables us to extend desirability to the nonlinear case by letting the utility scale be represented via a general closure operator. The new they directly expresses rewards in actual nonlinear currency (money), much in Savage's spirit, while arguably weakening the founding assumptions to a minimum. We characterise the main properties of the new theory both from the perspective of sets of gambles and of their lower and upper prices (previsions). We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.

Annals of Operations Research, in press.

# ENRIQUE MIRANDA, IGNACIO MONTES, ANDRÉS PRESA. INNER APPROXIMATIONS OF COHERENT LOWER PROBABILITIES AND THEIR APPLICATION TO DECISION MAKING PROBLEMS.

We consider a decision making problem under imprecision, where the probabilistic information is given in terms of a set of probability measures, and where finding the optimal alternative(s) may be difficult. To ease the computation, we propose to transform the initial model into another one that (i) belongs to some subclass with better mathematical properties, such as supermodularity or complete monotonicity; (ii) is at least as informative as the original model, while being as close as possible to it. We show that the problem can be approached in terms of linear or quadratic programming and that it can be connected with the one of determining the incenter of a credal set. Finally, we compare the solutions of a decision making problem with the initial and the transformed models and illustrate how our approach can be applied in a decision making problem under severe uncertainty.

International Journal of Approximate Reasoning, 167, 109132, 2024.

# ENRIQUE MIRANDA, RENATO PELESSONI, PAOLO VICIG. EVALUATING UNCERTAINTY WITH VERTICAL BARRIER MODELS.

Vertical Barrier Models (VBM) are a family of imprecise probability models that generalise a number of well known distortion/neighbourhood models (such as the Pari-Mutuel Model, the Linear-Vacuous Model, and others) while still being relatively simple. Several of their properties were established in previous works; in this paper we explore, in a finite framework, further facets of these models: their interpretation as neighbourhood models, the structure of their credal set in terms of maximum number of its extreme points, the result of merging operations with VBMs, the properties of their mass function, the conditions for VBMs to be belief functions or maxitive measures and the approximation of other models by VBMs.