1,349 research outputs found
Independent natural extension for sets of desirable gambles
We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal sets of gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence
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
Exchangeability and sets of desirable gambles
Sets of desirable gambles constitute a quite general type of uncertainty
model with an interesting geometrical interpretation. We give a general
discussion of such models and their rationality criteria. We study
exchangeability assessments for them, and prove counterparts of de Finetti's
finite and infinite representation theorems. We show that the finite
representation in terms of count vectors has a very nice geometrical
interpretation, and that the representation in terms of frequency vectors is
tied up with multivariate Bernstein (basis) polynomials. We also lay bare the
relationships between the representations of updated exchangeable models, and
discuss conservative inference (natural extension) under exchangeability and
the extension of exchangeable sequences.Comment: 40 page
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
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
Marginal extension in the theory of coherent lower previsions
AbstractWe generalise Walley’s Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the so-called forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions
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