Filip Hermans

Imprecise markov chains and their limit behavior

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithms operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.

Imprecise probability trees: Bridging two theories of imprecise probability

We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walleys behavioural theory of imprecise probabilities, and Shafer and Vovks game-theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walleys theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers.

Accept & reject statement-based uncertainty models

We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss-as a bridge to existing frameworks-a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements. We develop a framework for modelling and reasoning based on a pair of gamble sets.We translate the framework to one in terms of a pair of partial preference orders.Our framework generalises most frameworks based on a linear utility assumption.Particularly, it generalises the theories of coherent linear and lower previsions.We show the frameworks usefulness with a symmetry judgements modelling example.

Characterisation of ergodic upper transition operators

We study ergodicity for upper transition operators: bounded, sub-additive and non-negatively homogeneous transformations of finite-dimensional linear spaces. Ergodicity provides a necessary and sufficient condition for Perron-Frobenius-like convergence behaviour for upper transition operators. It can also be characterised alternatively: (i) using a coefficient of ergodicity, and (ii) using accessibility relations. The latter characterisation states that ergodicity is equivalent with there being a single maximal communication (or top) class that is moreover regular and absorbing. We present an algorithm for checking these conditions that is linear in the dimension of the state space for the number of evaluations of the upper transition operator.

Ergodicity Conditions for Upper Transition Operators

We study ergodicity for upper transition operators: bounded, sub-additive and non-negatively homogeneous transformations of finite-dimensional linear spaces. Ergodicity provides a necessary and sufficient condition for Perron–Frobenius-like convergence behaviour for upper transition operators. It can also be characterised alternatively using accessibility relations: ergodicity is equivalent with there being a single maximal communication (or top) class that is moreover regular and absorbing. We present efficient algorithms for checking these conditions.