Research

I'm currently a CIRAD (French Agricultural Research Centre Working for International Development) research engineer working in Montpellier, in the IATE team (Agropolymer Engineering and Emerging Technologies). Although I'm interested in many aspects of uncertainty theories (including decision theory, graphical models, statistical inference, ...) and of applied mathematics (interval computations, operational research, ...), I'm currently focusing my research on the following subjects :

• Practical representations of uncertainty
• Information fusion
• Dependence/independence notions
• Propagation of uncertainty through mathematical models (including reverse propagation)
• Sensitivity analysis in presence of imprecision
• Knowledge discovery, (supervised) learning methods, classification
  

Information, uncertainty and imprecision are part of our everyday decisions (just think how you park your car, how you estimate the duration of a trip,... ). As long as these decisions do not involve too high stakes and do not concern too much people, informal processing is sufficient (One seldom run complex algorithms to decide wether he should take a car, a bike or a bus to go to work). But, if stakes are high and system complex, decisions have to be justified and formal processing must be used to treat information, uncertainty and imprecision. This is a major goal of uncertainty theories : dealing with available information (with a minimal amount of added assumptions) to take decision. Indeed, at the end, most of the problems end up with a decision to take.

Probability theory is undoubtedly the oldest theory of uncertainty and is of major importance. Nevertheless, it can be argued that classical probabilities are too precise to model scarce, unreliable or imprecise information, and that in this latter case imprecision or lack of knowledge have to be modeled by another means. Actually, many arguments indicate that ignorance can't be properly modeled by classical probabilities.

Recent years have witnessed an increasing interest in other uncertainty theories that are able to faithfully deal with both imprecision and uncertainty. Although other propositions exist, three main theories have emerged as the main candidates, not to replace, but to complement probability theory : possibility theory, evidence theory and imprecise probability theory.

Possibility Theory

Although possibility theory has emerged from fuzzy logic, and that possibility distributions are formally equivalent to fuzzy sets, ideas, applications and interpretations of possibility theory are very far from being a simple by-product of fuzzy logic. Its more interesting side is with little doubt its qualitative aspects, allowing one to deal with purely ordinal considerations without any need of numerical evaluations. On the other side, quantitative possibility theory offers a simple framework which is computationally convenient and has a very intuitive interpretation in term of nested confidence interval. Quantative possibility theory has two main interpretations : it can considered as a direct extension of interval computation, or as the simplest model of a family of probability.  Of course, simplicity often means less expressive power, and this is the case here. So, one can wish to deal with more expressive theories, such as evidence theory or imprecise probability theory.

Evidence Theory

Evidence theory is often quoted as Dempster-Shafer theory, although it can have different interpretations, depending if it is considered as a special case of imprecise probabilities or as a model by itself. Dempster's view is more in agreement with the former case, while Shafer's original interpretation is related to the latter. This last interpretation has retain the attention of Philippe Smets, who used it as a basis for his Transferable Belief Model. Although many formal results and operations are the same in the two interpretations, Dempster's model and the TBM both rely on very different axiomatic. While Dempster model consists of a multi-valued mapping from a probability space to another space, TBM rely on three main different axioms : An open world assumption (allowing for unknown state of the world), a credal state, where evidence are entertained, discounted and revised and a pignistic state where decision is finally made (transforming the belief model into a pignistic probability).

Imprecise probabilities

Roughly speaking, imprecise probabilities consider sets of probability distributions instead of a single probability distribution, allowing thus uncertainty evaluation  to be imprecise. This theory, mainly developped by Peter Walley,  has many similarities with the subjective interpretation of probability developed by Bruno De Finetti (but is far from a mere extension of this theory). Provided one accepts Walley's behavioural interpretation of imprecise probabilities, this theory has a nice unifying feature, since both possibility theory and evidence theory can then be seen as special cases of imprecise probability. This theory has a very high expressive power and allows for a lot of flexibility, but this expressiveness is often paid by an higher computational complexity (The issue of making imprecise probabilistic model, being very important in practical applications, is considered by numerous authors).