Irrelevant natural extension for choice functions

We consider coherent choice functions under the recent axiomatisation proposed by De Bock and De Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment

Exchangeable choice functions

We investigate how to model exchangeability with choice functions. Exchangeability is a structural assessment on a sequence of uncertain variables. We show how such assessments are a special indifference assessment, and how that leads to a counterpart of de Finetti's Representation Theorem, both in a finite and a countable context

Learning imprecise hidden Markov models

We present a method for learning imprecise local uncertainty models in stationary hidden Markov models. If there is enough data to justify precise local uncertainty models, then existing learning algorithms, such as the Baum–Welch algorithm, can be used. When there is not enough evidence to justify precise models, the method we suggest here has a number of interesting features

Recent advances in imprecise-probabilistic graphical models

We summarise and provide pointers to recent advances in inference and identification for specific types of probabilistic graphical models using imprecise probabilities. Robust inferences can be made in so-called credal networks when the local models attached to their nodes are imprecisely specified as conditional lower previsions, by using exact algorithms whose complexity is comparable to that for the precise-probabilistic counterparts

Choice functions as a tool to model uncertainty

Our aim is to develop a tool for modelling different types of assessments about the uncertain value of some random variable. One well-know and widely used way to model uncertainty is using probability mass functions. However, such probability mass functions are not general enough to model, for instance, a total lack of knowledge. A very successful tool for modelling more general types of assessments is coherent sets of desirable gambles. These have many applications in credal networks, predictive inference, conservative reasoning, and so on. However, they are not capable of modelling beliefs corresponding to 'or' statements, for example the belief that a coin has two equal sides of unknown type: either twice heads or twice tails. Such more general assessments can be modelled with coherent choice functions. The first thing we do is relate coherent choice functions to coherent sets of desirable gambles, which yields an expression for the most conservative coherent choice function compatible with a coherent set of desirable gambles. Next, we study the order-theoretic properties of coherent choice functions. In order for our theory of choice functions to be successful, we need a good conditioning rule. We propose a very intuitive one, and show that it coincides with the usual one for coherent sets of desirable gambles, and therefore also leads to Bayes’s rule. To conclude, we show how to elegantly deal with assessments of indifference

Connecting choice functions and sets of desirable gambles

We study Seidenfeld, Schervish, and Kadane’s notion of choice functions, and want to make them accessible to people who are familiar with sets of desirable gambles. We relate both theories explicitly using their derived strict partial orderings. We give an expression for the most conservative extension of a set of desirable gambles to a choice function. Because it is important for inference purposes, we also make a link with belief structures