This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of

A credal network associates sets of probability distributions with directed acyclic graphs. Under strong independence assumptions, inference with credal networks is equivalent to a signomial program under linear constraints, a problem that is NP-hard even for categorical variables and polytree models. We describe

This paper reviews algorithms for local computation with imprecise probabilities. These algorithms try to solve problems of inference (calculation of conditional or unconditional probabilities) in cases in which there are a large number of variables. There are two main types depending on the nature of assumed independence relationships in each case. In both of them the global knowledge is composed of several pieces of local information. The objective is to carry out a sound global computation but mainly using the initial local representation. Keywords. Propagation algorithms, valuations based systems, imprecise probabilities. 1

Kuznetsov's condition says that variables X and Y are independent when any product of bounded functions f (X) and g(Y ) behaves in a certain way: the interval of expected values f (X)g(Y )] must be equal to the interval product f (X)]E[g(Y)]. The main result of this paper shows how to compute lower expectations using Kuznetsov's condition. We also generalize Kuznetsov's condition to conditional expectation intervals, and study the relationship between Kuznetsov's conditional condition and the semi-graphoid properties.

This paper analyzes independence concepts for sets of probability measures associated with directed acyclic graphs. The paper shows that epistemic independence and the standard Markov condition violate desirable separation properties. The adoption of a contraction condition leads to d-separation but still fails to guarantee a belief separation property. To overcome this unsatisfactory situation, a strong Markov condition is proposed, based on epistemic independence. The main result is that the strong Markov condition leads to strong independence and does enforce separation properties; this result implies that (1) separation properties of Bayesian networks do extend to epistemic independence and sets of probability measures, and (2) strong independence has a clear justi- cation based on epistemic independence and the strong Markov condition. 1

We apply random set theory to an analysis of future climate change. Bounds on cumulative probability are used to quantify uncertainties in natural and socio-economic factors that influence estimates of global mean temperature. We explore the link of...

Bayesian network are powerful probabilistic graphical models for modelling uncertainty. Among others, classification represents an important application: some of the most used classifiers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided for quantification. This requirement is sometimes too narrow. Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions. This leads to a novel class of imprecise probabilistic graphical models, called credal networks. In particular, classifiers based on Bayesian networks are generalized to so-called credal classifiers. Unlike Bayesian classifiers, which always detect a single class as the one maximizing the posterior class probability, a credal classifier may eventually be unable to discriminate a single class. In other words, if the available information is not sufficient, credal classifiers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian classifiers.

A second-order hierarchical uncertainty model of a system of independent random variables is studied in the paper. It is shown that the complex nonlinear optimization problem for reducing the second-order model to the firstorder one can be represented as a finite set of simple linear programming problems with a finite number of constraints. The stress-strength reliability analysis by unreliable information about statistical parameters of the stress and strength exemplifies the model. Numerical examples illustrate the proposed algorithm for computing the stress-strength reliability.

This paper discusses concepts of independence and their relationship with convexity assumptions in the theory of sets of probability distributions. The paper offers an organized review of the literature and some new ideas (on regular conditional independence and exchangeability/“strong independence”). Finally, the connection between recent developments on the axiomatization of non-binary preferences, and its impact on “strict” independence, are analyzed.

The computational manipulation of probability measures often requires the treatment of interval values, not only due to numerical errors, but also due to more fundamental difficulties: we may want to model imprecise beliefs; we may have incomplete knowledge about probability values; we may be interested in merging beliefs from groups of experts; and we may wish to verify the effect of perturbations in probabilistic models [2, 17]. Such difficulties have often led to the study of interval probability and related theories. The goal of this paper is to present a brief overview of methods and results that can be relevant to the validated manipulation of probabilistic models. The most general representation for imprecision in probabilistic models seems to be provided by the theory of sets of probabilities (called credal sets [13]). In this work we focus on closed convex credal sets; there are axiomatic derivations of such credal sets and other variants [12, 16, 17]. Consider two examples. First, consider a binary variable X and the set of measures defined by the interval P (X = x0) ∈ [0.3, 0.4], where P(X = x0) is the probability of the event