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Enrique Miranda

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9 papers

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ISIPTA Conference 2025 Conference Paper

Distorting lower probabilities using common distortion models

David Nieto-Barba
Ignacio Montes
Enrique Miranda

Distortion or neighbourhood models are useful tools in the imprecise probability theory allowing to robustify a probabilistic model by considering a neighbourhood around a given probability measure. In this work, we tackle the more general problem of distorting a lower probability. This problem can be interesting when we believe that a given lower probability is too precise, or in coalitional game theory when the set of solutions is empty. Our main purpose is to investigate how the linear vacuous and pari mutuel models can be defined for the distortion of lower probabilities, and for this aim we address the problem in a more general manner: we extend the vertical barrier models, which include the linear vacuous and pari mutuel models, and investigate the properties they satisfy.

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ISIPTA Conference 2025 Conference Paper

On the closure of aggregation rules for imprecise probabilities

Enrique Miranda
Ignacio Montes

We consider the problem of aggregating a number of imprecise probability models into a joint one, and compare four aggregation rules: conjunction, disjunction, mixture and Pareto. We investigate for which particular cases of imprecise probability models these operators are closed, meaning that the output belongs to the same family as the inputs. Specifically, we analyse this problem for comparative probability models, $2$-monotone capacities, probability intervals, belief functions, p-boxes and minitive measures.

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AIJ Journal 2020 Journal Article

Compatibility, desirability, and the running intersection property

Enrique Miranda
Marco Zaffalon

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.

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ISIPTA Conference 2017 Conference Paper

A Study of the Pari-Mutuel Model from the Point of View of Imprecise Probabilities

Ignacio Montes
Enrique Miranda
Sébastien Destercke

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

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JAIR Journal 2017 Journal Article

Axiomatising Incomplete Preferences through Sets of Desirable Gambles

Marco Zaffalon
Enrique Miranda

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.

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ISIPTA Conference 2017 Conference Paper

Game Solutions, Probability Transformations and the Core

Enrique Miranda
Ignacio Montes

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

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AIJ Journal 2013 Journal Article

Probability and time

Marco Zaffalon
Enrique Miranda

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 approach 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.

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AIJ Journal 2011 Journal Article

Independent natural extension

Gert de Cooman
Enrique Miranda
Marco Zaffalon

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. This gives epistemic independence very wide scope as well as appeal: this interpretation of independence is often taken as natural also in precise-probabilistic contexts. Nevertheless, epistemic independence has received little attention so far. This paper develops the foundations of this notion 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, such as associativity, marginalisation and 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 judgements. 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.

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AIJ Journal 2009 Journal Article

Coherence graphs

Enrique Miranda
Marco Zaffalon

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.

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