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 nontrivial generalization of the classical Perron-Frobenius theorem to imprecise Markov chains.
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 algorithm's 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.
We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's 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 Walley's 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.
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. OverviewAs stated, the prime goal of this paper is to present a mathematical framework for modelling uncertainty. The basic conceptual set-up and mathematical notation is given in Section 1.4. Our presentation starts from the basics and there is no prerequisite knowledge of (imprecise) probability theory, although of course this does not hurt. However, throughout, a basic familiarity with set theory, linear algebra, and convex analysis is assumed. Once the framework is established in Section 2, we do make the connection with preference relations in Section 3 and (imprecise) probability frameworks in Sections 4 and 5. Also here specific prerequisite knowledge is useful but not necessary. Before concluding with some final remarks, musings and topics for further investigation in Section 7, we present a small theoretical application of our framework in Section 6, i.e., we show how it can be effectively used to model symmetry judgements. To improve the readability of the paper, we have gathered the proofs in an appendix.How do we build up our framework? We start by giving a detailed description of the nature of the assessments we consider, the pairs of sets of accepted and rejected gambles (Section 2.1). In three subsequent sections, we introduce the main rationality criteria of our framework: No Confusion, which determines the assessments that are irrational (Section 2.2); Deductive Closure, which makes explicit the consequences of assuming values are expressed in a linear precise utility scale (Section 2.4); No Limbo, which states that gambles whose acceptance would lead to an irrational model must be rejected. The latter two axioms are accompanied by operators that extend an assessment to a full fledged model that incorporates both the assessments and the requirements of these axioms. At this point the conceptual groundwork is complete.The central question then becomes: given an assessment, what are the properties of the model obtained by extending it, i.e., by performing deductive inference? In particular: which assessments give rise to rational models? To be able to provide general answers to these questions, we first perform an order-theoretic analysis of the sets of all assessments and models (Section 2.6), where assessments are ordered according to the resolve they encode, i.e., the amount of accepted and rejected gambles they contain. Both general answers are given (in Section 2.7) and answers for the important specif...
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