http://www.sipta.org/ipp.xml If you want to post an abstract, just send a message to alessandro@idsia.ch with the necessary information http://www.hds.utc.fr/~tdenoeux/perso/doku.php?id=en:publi:belief_art T. Denoeux, Z. Younes and F. Abdallah A formalism is proposed for representing uncertain information on set-valued variables using the formalism of belief functions. A set-valued variable $X$ on a domain $\Omega$ is a variable taking zero, one or several values in $\Omega$. While defining mass functions on the frame $2^{2^\Omega}$ is usually not feasible because of the double-exponential complexity involved, we propose an approach based on a definition of a restricted family of subsets of $2^\Omega$ that is closed under intersection and has a lattice structure. Using recent results about belief functions on lattices, we show that most notions from Dempster-Shafer theory can be transposed to that particular lattice, making it possible to express rich knowledge about $X$ with only limited additional complexity as compared to the single-valued case. An application to multi-label classification (in which each learning instance can belong to several classes simultaneously) is demonstrated. http://bellman.ciencias.uniovi.es/~emiranda/regular-extension.pdf E. Miranda and M. Zaffalon 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 sequen ces 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. http://cms.brookes.ac.uk/staff/FabioCuzzolin/pubs.html Fabio Cuzzolin In this paper we provide a comprehensive study of several of the most popular Bayesian approximations of a belief function in the probability simplex. Starting from the interpretation of the pignistic function as center of mass of the simplex of consistent probabilities, we prove that a large group of approximations can be described by the notion of focus of pairs of simplices in the simplex of all probability measures. http://cms.brookes.ac.uk/staff/FabioCuzzolin/pubs.html Fabio Cuzzolin In this paper we introduce three alternative combinatorial formulations of the theory of evidence (ToE), by proving that both plausibility and commonality functions share the structure of sum function with belief functions. We compute their Moebius inverses, which we call basic plausibility and commonality assignments. As these results are achieved in the framework of the geometric approach to uncertainty measures, the equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. We can then describe the point-wise geometry of these sum functions in terms of rigid transformations mapping them onto each other. Combination rules can be applied to plausibility and commonality functions through their Moebius inverses, leading to interesting applications of such inverses to the probabilistic transformation problem. http://cms.brookes.ac.uk/staff/FabioCuzzolin/pubs.html Fabio Cuzzolin In this paper we extend the geometric approach to the theory of evidence in order to include other important ¯nite fuzzy measures. In particular we describe the geometric counterparts of the class of possibility measures represented by consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex. In particular we focus on outer consonant approximations, showing that they live on a polytopes associated with all possible maximal chains of focal elements, which in turn form a simplicial complex in analogy with the whole consonant space. http://sdestercke.free.fr/papers/PossAppRSI_IJUFKS_DesterckeDuboisChoj.pdf S. Destercke, D. Dubois and E. Chojnacki The belief structure resulting from the combination of consonant and independent marginal random sets is not, in general, consonant. Also, the complexity of such a structure grows exponentially with the number of combined random sets, making it quickly intractable for computations. In this paper, we propose a simple guaranteed consonant outer approximation of this structure. The complexity of this outer approximation only linearly increases with the number of marginal random sets (i.e., of dimensions), making it easier to handle in uncertainty propagation. Features and advantages of this outer approximation are then discussed, with the help of some illustrative examples. http://isas.uka.de/Publikationen/Fusion09_Klumpp-Type2Dens.pdf Vesa Klumpp, Uwe D. Hanebeck In this paper, we address the problem of processing imprecisely known probability density functions by means of Bayesian estimation. The imprecise knowledge about probability density functions is given as stochastic uncertainty about their parameters. The proposed processing of this special density in a Bayesian estimator is accomplished by reinterpretation of the Filter and prediction equations. Here, the parameters are treated as a higher order state, which can be processed by Bayesian estimation techniques. For state estimation, this avoids the need to select specific values for unknown parameters and, thus, allows the processing of all potential parameters at once. The proposed approach further allows the use of imprecisely known model equations for measurement and state prediction by the same principle. http://isas.uka.de/Publikationen/Fusion09_Noack.pdf Benjamin Noack, Vesa Klumpp, Uwe D. Hanebeck In practical applications, state estimation requires the consideration of stochastic and systematic errors. If both error types are present, an exact probabilistic description of the state estimate is not possible, so that common Bayesian estimators have to be questioned. This paper introduces a theoretical concept, which allows for incorporating unknown but bounded errors into a Bayesian inference scheme by utilizing sets of densities. In order to derive a tractable estimator, the Kalman filter is applied to ellipsoidal sets of means, which are used to bound additive systematic errors. Also, an extension to nonlinear system and observation models with ellipsoidal error bounds is presented. The derived estimator is motivated by means of two example applications. http://isas.uka.de/Publikationen/Fusion08_Noack.pdf Benjamin Noack, Vesa Klumpp, Dietrich Brunn, Uwe D. Hanebeck This paper presents a theoretical framework for Bayesian estimation in the case of imprecisely known probability density functions. The lack of knowledge about the true density functions is represented by sets of densities. A formal Bayesian estimator for these sets is introduced, which is intractable for infinite sets. To obtain a tractable filter, properties of convex sets in form of convex polytopes of densities are investigated. It is shown that pathwise connected sets and their convex hulls describe the same ignorance. Thus, an exact algorithm is derived, which only needs to process the hull, delivering tractable results in the case of a proper parametrization. Since the estimator delivers a convex hull of densities as output, the theoretical grounds are laid for deriving efficient Bayesian estimators for sets of densities. The derived filter is illustrated by means of an example. http://www.idsia.ch/~alessio/hmm-short_rev.pdf Benavoli, A., Zaffalon, M., Miranda, E. We extend Hidden Markov Models for continuous variables taking into account imprecision in our knowledge about the probabilistic relationships involved. To achieve that, we consider sets of probabilities, also called coherent lower previsions. In addition to the general formulation, we study in detail a particular case of interest: linear-vacuous mixtures. We also show, in a practical case, that our extension outperforms the Kalman filter when modelling errors are present in the system. http://www.idsia.ch/~alessandro/papers/antonucci2009e.pdf Antonucci, A., Benavoli, A., Zaffalon, M., de Cooman, G., Hermans, F. We present a new procedure for tracking manoeuvring objects by hidden Markov chains. It leads to more reliable modelling of the transitions between hidden states compared to similar approaches proposed within the Bayesian framework: we adopt convex sets of probability mass functions rather than single ''precise probability'' specifications, in order to provide a more realistic and cautious model of the manoeuvre dynamics. In general, the downside of such increased freedom in the modelling phase is a higher inferential complexity. However, the simple topology of hidden Markov chains allows for efficient tracking of the object through a recently developed belief propagation algorithm. Furthermore, the imprecise specification of the transitions can produce so-called indecision, meaning that more than one model may be suggested by our method as a possible explanation of the target kinematics. In summary, our approach leads to a multiple-model estimator whose performance, investigated through extensive numerical tests, turns out to be more accurate and robust than that of Bayesian ones.