Enrique Miranda

Marginal extension in the theory of coherent lower previsions

We generalise Walleys 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.

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. 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.

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 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.

Approximations 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.

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 random variable. 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.

Robust Filtering Through Coherent Lower Previsions

The classical filtering problem is re-examined to take into account imprecision in the knowledge about the probabilistic relationships involved. Imprecision is modeled in this paper by closed convex sets of probabilities. We derive a solution of the state estimation problem under such a framework that is very general: it can deal with any closed convex set of probability distributions used to characterize uncertainty in the prior, likelihood, and state transition models. This is made possible by formulating the theory directly in terms of coherent lower previsions, that is, of the lower envelopes of the expectations obtained from the set of distributions. The general solution is specialized to two particular classes of coherent lower previsions. The first consists of a family of Gaussian distributions whose means are only known to belong to an interval. The second is the so-called linear-vacuous mixture model, which is a family made of convex combinations of a known nominal distribution (e.g., a Gaussian) with arbitrary distributions. For the latter case, we empirically compare the proposed estimator with the Kalman filter. This shows that our solution is more robust to the presence of modelling errors in the system and that, hence, appears to be a more realistic approach than the Kalman filter in such a case.

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.

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.

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.

A random set characterization 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 Mobius 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.