# Research

My early work (related to my doctoral research) focused on a special type of uncertainty models, called *possibility measures*, which are special, maxitive, set functions that emerge as models for linguistic uncertainty in the context of L. Zadeh's fuzzy set theory. They were also used earlier by G. L. S. Shackle as *measures of potential surprise*.

I fairly quickly became dissatisfied with the lack of a good operationalisable semantics and interpretation for fuzzy sets and the related theory of possibility measures. Reading Peter Walley's *Statistical Reasoning with Imprecise Probabilities*, convinced me that possibility measures had a role to play in the behavioural theory of imprecise probabilities. This led to a second phase in my research, in which I investigated to what extent possibility measures, and other notions from possibility theory (and even to some extent from fuzzy set theory) could be given a behavioural interpretation, and embedded into behavioural theory of imprecise probabilities.

Since then, my research has moved firmly into the domain of imprecise probabilities itself. Indeed, my current main research interest lies in *models for the representation and manipulation of uncertainty*, using the so-called *behavioural theory of imprecise probabilities*.

I am convinced that specifying a precise probability model (a probability measure, or a prevision) in order to model uncertain knowledge is often unrealistic. Probability measures represent very strong information states, and their use are not always warranted by the amount of information that is actually available. We therefore need tools to reason with information states that are less than ideal, or less than perfect. Imprecise probability models aim not to replace, but to complement and enlarge the classical, or Bayesian, approach to probability theory, by providing it with tools to work with weaker information states.

See the website of the International Society for Imprecise Probability: Theories and Applications (SIPTA) for more information about this theory.

My work in this area is centred around the following topics:

- Mathematical aspects of imprecise probability models (coherence, independence, conditioning, natural extension, integration,
*n*-monotonicity) - Sets of desirable gambles as a foundation for uncertain reasoning
- Choice functions as a foundation for uncertain reasoning
- Incorporating possibility theory in the theory of imprecise probabilities
- Belief change with general information states
- Imprecise hierarchical uncertainty models
- Learning and optimisation using imprecise probability models, with applications in systems modelling, identification and control, classification, ...
- Imprecise probability models for dealing with missing data
- Symmetry in imprecise probability models, and in particular issues of exchangeability
- Laws of large numbers and ergodic theorems for imprecise probability models
- Predictive inference and the Imprecise Dirichlet Model
- Applications of the Imprecise Dirichlet Model in game theory, (hidden) Markov models, ...
- Connections between different types of imprecise probability models
- Game-theoretic probability and uncertain/random processes with imprecise probability, in particular imprecise probability trees, and Markov chains with imprecise transition probabilities
- Martingale theoretic approaches to imprecise probabilities
- Epistemic irrelevance as an `independence' notion in credal networks, and efficient algorithms for making inferences in credal nets under epistemic irrelevance