Invited Speakers

Reinhart Viertl

"Fuzzy Bayesian Inference"

Fuzziness of data is usually neglected is statistics. But real data are frequently not precise numbers, but more or less imprecise. This imprecision is different from errors. Imprecision of data can be modelled by special fuzzy subsets of the set of real numbers, and statistical methods have to be generalized to fuzzy data.

Another kind of fuzziness is the fuzziness of a-priori information in Bayesian inference. It is possible to apply so-called fuzzy probability distributions as a-priori distributions. The corresponding generalization of Bayes´ theorem is basic for what is called fuzzy Bayesian inference.

Dominique Guyonnet

"Soft methods for treating uncertainties : applications in the field of environmental risks"

Theoretical developments that help account for epistemic uncertainties in risk analyses (imprecise probability theory, possibility theory, evidence theory, ...) are now being applied to estimate risks in the environmental field. Example applications exist in the fields of air, soil and water pollution. This paper presents an application to a new area; i.e. climate change mitigation technologies. Since the start of the industrial era, the average temperature of the Earth has increased by 0.6oC while the greenhouse gas content of the atmosphere has increased by approx. 50%. It is foreseen that by 2020-2030, fossil energy will still continue to satisfy 80% of the energetic demands of the world's increasing population. If current trends are not somehow mitigated, weather simulations suggest average temperature increases in excess of 2oC, with dramatic consequences for life on Earth. Carbon Capture and Storage (CCS) is increasingly identified as a major potential climate change mitigation technology, to the extent that a European Directive regarding this technology is currently being considered. The technology consists schematically in (i) capturing carbon dioxide at the production site (e.g. a fossil-fuel energy production plant), (ii) transporting it over more-or-less large distances to the injection site and (iii) injecting it into deep geological "trap" deposits. The feasibility and long-term efficiency of the deep injection must be assessed in the light of potential risks involved. Primary risks are (i) the potential for carbon dioxide leakage to the surface with impacts to human health and (ii) leakage to resource aquifers with modifications of aquifer chemistry and impacts on groundwater quality. BRGM is currently in the process of assessing these risks which are largely influenced by various types of uncertainties: stochastic uncertainties related to the natural heterogeneity of the subsurface and epistemic uncertainties due to the imprecise/incomplete nature of the knowledge base. The paper summarizes progress to-date in the risk assessments and presents first results of uncertainty propagation analyses. The emphasis is on the joint propagation of both types of uncertainties, using so-called "hybrid-type" methods that borrow from the general concept of imprecise probability. Another aspect that is examined is the synthesis and presentation of results for decision-makers. It is proposed to introduce "a posteriori" subjectivity at a decision-making stage, in the form of a "confidence index", as opposed to the "a priori" subjectivity proposed in a Bayesian framework, which introduces confusion between epistemic and stochastic uncertainties.

Jean-Marc Bernard

"Imprecise probabilistic prediction for categorical data: from Bayesian inference to the Imprecise Dirichlet-multinomial models (IDM & IDMM)"

From n observations, what can be predicted about some future n'observations? Suppose the data are categorical, i.e. each observation belongs to one amongst K exclusive categories. The observed data yield the counts a=(a_1,...,a_K) over the K categories, whose sum is equal to n, and we would like to make predictions about the unknown composition a'=(a'_1,...a'_K) of the next n' observations. Karl Pearson considered this as the "fundamental problem of statistical inference". The problem has a long history which can be traced back to the works of Bayes and Laplace. It is in fact within the Bayesian framework that the problem has been mostly addressed: in that framework, prior uncertainty, about a and a', and posterior uncertainty, about a' given a, are each described by a single probability distribution. Consider the case n'=1, and some event B which has been observed a_B times out of n. Then, the probability that B will also be observed for the next observation, P(B|a), is typically of the form P(B|a)=(a_B+\alpha)/(n+\alpha+\beta), where \alpha and \beta are positive scalars. For instance, in the case K=2 and n'=1, the famous Laplace's "rule of succession" corresponds to \alpha=\beta=1. It is obtained by the use of a Beta-binomial distribution as a prior, or more generally a Dirichlet-multinomial distribution for general K. Other Bayesian approaches to formalizing prior ignorance yield the same form for P(B|a) with different \alpha and \beta.

In this talk, we present a generalization of this Bayesian model, the imprecise Dirichlet-multinomial model (IDMM) (Walley, Bernard, 1999, Technical report), which uses sets of Dirichlet-multinomial distributions to model prior ignorance (instead of a single one), and yields lower and upper probabilities for events of interest. For the above example, P(B|a) still has the same form, but \alpha and \beta can take any positive values such that \alpha+\beta=s, where s is a constant. The IDMM can be viewed as the predictive part of the imprecise Dirichlet model (IDM) (Walley, 1996, JRSS B). The IDMM is a special case of a "coherent lower previsions" model. Hence, the IDMM has several properties of great interest which will be emphasized: it satisfies coherence, symmetry and some desirable invariance properties, which are generally not satisfied jointly by precise Bayesian models. Also, the IDMM encompasses various alternative precise models (Bayesian or frequentist) for quite small values of s. The talk will provide the opportunity to discuss several issues: formalization of prior ignorance, incorporating prior information, inference about a universal law, generality and uses of the predictive approach to inference, advantages of imprecise probability models to describe uncertainty.

IRIT, Université Paul Sabatier, Conseil Régional Midi-Pyrénées, Mairie de Toulouse, IRSN, CajAstur, EUSFLAT

October 2008