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

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

Credal networks under epistemic irrelevance using sets of desirable gambles

We present a new approach to credal networks, which are graphical models that generalise Bayesian nets to deal with imprecise probabilities. Instead of applying the commonly used notion of strong independence, we replace it by the weaker notion of epistemic irrelevance. We show how assessments of epistemic irrelevance allow us to construct a global model out of given local uncertainty models, leading to an intuitive expression for the so-called irrelevant natural extension of a network. In contrast with Cozman (2000), who introduced this notion in terms of credal sets, our main results are presented using the language of sets of desirable gambles. This has allowed us to derive a number of useful properties of the irrelevant natural extension. It has powerful marginalisation properties and satisfies all graphoid properties but symmetry, both in their direct and reverse forms

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

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

Empirical interpretation of imprecise probabilities

This paper investigates the possibility of a frequentist interpretation of imprecise probabilities, by generalizing the approach of Bernoulli’s Ars Conjectandi. That is, by studying, in the case of games of chance, under which assumptions imprecise probabilities can be satisfactorily estimated from data. In fact, estimability on the basis of finite amounts of data is a necessary condition for imprecise probabilities in order to have a clear empirical meaning. Unfortunately, imprecise probabilities can be estimated arbitrarily well from data only in very limited settings

Credal Networks under Epistemic Irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally. The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning