On coherent immediate prediction: connecting two theories of imprecise probability

We give an overview of two approaches to probabiliity 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 immediate prediction in the framework of Walley's theory, and we prove an interesting and quite general version of the weak law of large numbers

Modeling uncertainty using accept & reject statements

Uncertainty and preference is often modeled using linear previsions and linear orders. Some more expressive models use sets of probabilities, lower previsions, or partial orders (see, e.g., the work of Seidenfeld et al. and Walley). In the discussion of these more expressive models, or even to justify them, alternative representations in terms of sets of so-called acceptable, favorable, or desirable gambles appear (cf. the work of Williams, Seidenfeld et al., and Walley). Such ‘sets of gambles’-based models are attractive because of their geometric nature. We generalize these ‘sets of gambles’-based models by considering a pair of sets, one with accepted gambles and one with rejected gambles. We develop a framework based on a small number of axioms—No Confusion, Deductive Closure, No Limbo, and Indifference to Status Quo—and provide an interesting characterization of the resulting models. Furthermore, we define a pair of equivalent gamble relations that generalize the partial orders mentioned earlier; the corresponding characterization result is also given

Limit behaviour for imprecise Markov Chains

An imprecise Markov chain is defined by a closed convex set of transition matrices instead of a unique one for a classical precise Markov chain. These imprecise Markov chains allow us to model situations where we do not have enough information to specify a unique transition matrix, or to approximate the behaviour of non‐stationary Markov chains. We show that there are efficient, dynamic programming‐ like ways to work and reason with these imprecise Markov chains; e.g. to calculate the resulting distribution over the states at any time instant. We prove that this distribution converges in time, similarly to the precise case and under very mild conditions. We thus effectively prove a Perron‐Frobenius theorem for a special class of non‐linear systems

Belief propagation in imprecise Markov trees

We replace strong independence in credal networks with the weaker notion of epistemic irrelevance. Focusing on directed trees, we show how to combine the local credal sets in the networks into an overall joint model, and use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the network. The algorithm, which is essentially linear in the number of nodes, is formulated entirely in terms of coherent lower previsions. We supply examples of the algorithm's operation, and report an application to on-line character recognition that illustrates the advantages of the model for prediction

Propagating imprecise probabilities through event trees

Event trees are a graphical model of a set of possible situations and the possible paths going through them, from the initial situation to the terminal situations. With each situation, there is associated a local uncertainty model that represents beliefs about the next situation. The uncertainty models can be classical, precise probabilities; they can also be of a more general, imprecise probabilistic type, in which case they can be seen as sets of classical probabilities (yielding probability intervals). To work with such event trees, we must combine these local uncertainty models. We show this can be done efficiently by back-propagation through the tree, both for precise and imprecise probabilistic models, and we illustrate this using an imprecise probabilistic counterpart of the classical Markov chain. This allows us to perform a robustness analysis for Markov chains very efficiently

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 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.Comment: 29 pages, 5 figures, 1 tabl

Epistemic irrelevance in credal networks : the case of imprecise Markov trees

We replace strong independence in credal networks with the weaker notion of epistemic irrelevance. Focusing on directed trees, we show how to combine local credal sets 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 essentially linear in the number of nodes, is formulated entirely in terms of coherent lower previsions. We supply examples of the algorithm's operation, and report an application to on-line character recognition that illustrates the advantages of our model for prediction

Characterisation of ergodic upper transition operators

AbstractWe 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