Unifying Practical Uncertainty Representations: II. Clouds

There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models . This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by 2-monotone (convex) capacities.Comment: 30 pages, 7 figures, Pre-print of journal paper to be published in International Journal of Approximate Reasoning (with expanded section concerning clouds and probability intervals

Computing Expectations with Continuous P-Boxes: Univariate Case

Given an imprecise probabilistic model over a continuous space, computing lower/upper expectations is often computationally hard to achieve, even in simple cases. Because expectations are essential in decision making and risk analysis, tractable methods to compute them are crucial in many applications involving imprecise probabilistic models. We concentrate on p-boxes (a simple and popular model), and on the computation of lower expectations of non-monotone functions. This paper is devoted to the univariate case, that is where only one variable has uncertainty. We propose and compare two approaches : the first using general linear programming, and the second using the fact that p-boxes are special cases of random sets. We underline the complementarity of both approaches, as well as the differences.Comment: 31 pages, 6 figures, constitute an extended version of a small paper accepted in ISIPTA conference, and a preprint version of a paper accepted in IJA

EXTREME POINTS OF THE CREDAL SETS GENERATED BY COMPARATIVE PROBABILITIES

ABSTRACT. 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 mixtures 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 singletons of a finite space. We characterise these extreme points by mean of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses

Model checking for imprecise Markov chains.

We extend probabilistic computational tree logic for expressing properties of Markov chains to imprecise Markov chains, and provide an efficient algorithm for model checking of imprecise Markov chains. Thereby, we provide a formal framework to answer a very wide range of questions about imprecise Markov chains, in a systematic and computationally efficient way

A robust data driven approach to quantifying common-cause failure in power networks.

The standard alpha-factor model for common cause failure assumes symmetry, in that all components must have identical failure rates. In this paper, we generalise the alpha-factor model to deal with asymmetry, in order to apply the model to power networks, which are typically asymmetric. For parameter estimation, we propose a set of conjugate Dirichlet-Gamma priors, and we discuss how posterior bounds can be obtained. Finally, we demonstrate our methodology on a simple yet realistic example

A note on the temporal sure preference principle and the updating of lower previsions.

This paper reviews the temporal sure preference principle as a basis for inference over time. We reformulate the principle in terms of desirability, and explore its implications for lower previsions. We report some initial results. We also discuss some of the technical difficulties encountered

Logistic regression on Markov chains for crop rotation modelling.

Often, in dynamical systems, such as farmer's crop choices, the dynamics is driven by external non-stationary factors, such as rainfall, temperature, and economy. Such dynamics can be modelled by a non-stationary Markov chain, where the transition probabilities are logistic functions of such external factors. We investigate the problem of estimating the parameters of the logistic model from data, using conjugate analysis with a fairly broad class of priors, to accommodate scarcity of data and lack of strong prior expert opinions. We show how maximum likelihood methods can be used to get bounds on the posterior mode of the parameters

A Note on the Temporal Sure Preference Principle and the Updating of Lower Previsions

This paper reviews the temporal sure preference principle as a basis for inference over time. We reformulate the principle in terms of desirability, and explore its implications for lower previsions. We report some initial results. We also discuss some of the technical difficulties encountered

A Robust Data Driven Approach to Quantifying Common-Cause Failure in Power Networks

The standard alpha-factor model for common cause failure assumes symmetry, in that all components must have identical failure rates. In this paper, we generalise the alpha-factor model to deal with asymmetry, in order to apply the model to power networks, which are typically asymmetric. For parameter estimation, we propose a set of conjugate Dirichlet-Gamma priors, and we discuss how posterior bounds can be obtained. Finally, we demonstrate our methodology on a simple yet realistic example

Logistic Regression on Markov Chains for Crop Rotation Modelling

Often, in dynamical systems, such as farmer's crop choices, the dynamics is driven by external non-stationary factors, such as rainfall, temperature, and economy. Such dynamics can be modelled by a non-stationary Markov chain, where the transition probabilities are logistic functions of such external factors. We investigate the problem of estimating the parameters of the logistic model from data, using conjugate analysis with a fairly broad class of priors, to accommodate scarcity of data and lack of strong prior expert opinions. We show how maximum likelihood methods can be used to get bounds on the posterior mode of the parameters