## Research interests

My research is concerned with the theoretical study of imprecise probability models, and with the design of efficient algorithms for statistical inference and decision making with such models.

Simply put, imprecise probability models are just sets of probability models, the elements of which are typically regarded as candidates for some ideal true probability model. In this sense, imprecise probability theory is an extension of probability theory that combines probabilistic uncertainty and model uncertainty into a single theory. However, this simplified view is only one of the many ways to look at or interpret imprecise probabilities. Among the many other imprecise-probabilistic frameworks that exist, I favour lower previsions, sets of desirable gambles and choice functions.

From an applied point of view, imprecise probability models are useful whenever sharp, precise and trustworthy probabilities are not available, for example because it is too difficult, costly, or time-consuming to compute them, learn them from data or elicit them from experts. In such cases, standard probability models, and the inferences and decisions that are obtained by them, are often unreliable. Imprecise probability models avoid this problem by allowing for partial probability assessments such as bounds on probabilities. The resulting inferences and decisions are robust with respect to variations within these bounds, and will therefore remain reliable even in cases of severe uncertainty.

### Credal networks

My early work on imprecise probabilities was mainly concerned with credal networks, which are probabilistic graphical models that generalise Bayesian networks to the framework of imprecise probabilities, including special cases such as hidden Markov models and Markov chains. Basically, the main difference is that the local uncertainty models of a credal network are allowed to be imprecise, whereas a Bayesian network only considers probability distributions. This extra feature allows for more flexible modelling, and results in inferences and decisions that are more reliable. However, it also comes at a price: computing inferences in credal networks tends to be very complex, and efficient algorithms typically apply to specific inference problems only. Much of my early research aimed at reducing this complexity, by developing efficient algorithms for new types of inference problems.

In the context of my PhD, I have focussed on one particular type of credal networks, called credal networks under epistemic irrelevance. I have studied their theoretical properties, and have used these properties to design algorithms for them that can efficiently solve large classes of inference problems. Interestingly, these classes include problems that are known to be NP-hard for other types of credal networks, such as credal networks under strong or complete independence. After my PhD, I also designed some algorithms for these other types of credal networks. One of these algorithms, which I am particularly proud of, can be used to efficiently evaluate (and quantify) the robustness of individual MAP inferences in a large class of probabilistic graphical models, including Bayesian networks and Markov random fields.

### Imprecise stochastic processes and Markov chains

While I remain very much interested in credal networks, the focus of my research has by now largely shifted to other topics. One of these topics is stochastic processes and, in particular, their extension to the framework of imprecise probabilities. Initially, because of their direct link with credal networks, I focussed on imprecise versions of discrete-time Markov chains. Since then, my interest has grown to include various other types of imprecise stochastic processes as well, including imprecise Poisson processes, imprecise Dirichlet processes, imprecise multinomial processes and, most importantly, imprecise continuous-time Markov chains.

My research on imprecise stochastic processes includes foundational topics (exchangeability, convergence to limit distributions, ergodicity, game-theoretic stochastic processes, continuity properties), computational aspects such as the development of efficient inference methods, and their applications to statistical inference and queueing theory. One particular set of results that should be very useful to practitioners are our results on lumping, where we show that imprecise Markov chains can be used to deal with the curse of dimensionality that their precise counterparts often suffer from. That is, for any given Markov chain, if the size of its state space makes inference computationally infeasible, one can use an imprecise Markov chain with a smaller state space to efficiently compute guaranteed bounds on various inferences of interest.

### Choice functions and sets of desirable option sets

A second topic to which my research has shifted, and the one I currently work on the most, is choice functions. Quite simply, these are functions that choose: given any set of options, they choose among these options. Traditionally, a single option is typically chosen; for example, the option that maximises the expected utility with respect to some probability model. In the context of imprecise probabilities, however, the resulting choice may become imprecise as well, leading to set-valued choices. The practical aim of a choice function then, precise or imprecise, is to make decisions under uncertainty. Usually, this is achieved by combining an uncertainty model with a decision rule.

My theoretical research on this topic is mainly concerned with the development of a unifying mathematical framework for studying choice functions. The key insight from which this framework has grown, is that choice functions can be given an interpretation in terms of sets of desirable gambles. In particular, they are mathematically equivalent to so-called sets of desirable option sets. This insight has opened up a fascinating connection between choice functions and imprecise uncertainty models of all sorts, the extent of which I continue to explore, mainly in collaboration with Gert de Cooman. Among other things, we have developed conservative inference techniques for choice functions, and an axiomatic basis for the combination of particular uncertainty models and decision rules. My applied research on choice functions is concerned with using such combinations to robustify traditional classification techniques.

### Other topics

Besides on choice functions, imprecise stochastic processes and credal networks, I also work on, or have worked on and remain interested in, various other topics within the realm of imprecise probabilities. The following list provides a brief overview:

- Imprecise randomness (deriving imprecise models from infinite data sequences)
- Imprecise statistics (learning imprecise models from finite data sequences)
- Imprecise extensions of the Viterbi algorithm
- Imprecise extensions of optimal control algorithms
- Conditioning on events with (lower) probability zero
- Extreme lower previsions
- Sensitivity analysis

“La théorie des probabilités n’est, au fond, que le bon sens réduit au calcul;

elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d'instinct,

sans qu'ils puissent souvent s'en rendre compte.”

– Pierre-Simon Laplace