244 research outputs found
2-coherent and 2-convex Conditional Lower Previsions
In this paper we explore relaxations of (Williams) coherent and convex
conditional previsions that form the families of -coherent and -convex
conditional previsions, at the varying of . We investigate which such
previsions are the most general one may reasonably consider, suggesting
(centered) -convex or, if positive homogeneity and conjugacy is needed,
-coherent lower previsions. Basic properties of these previsions are
studied. In particular, we prove that they satisfy the Generalized Bayes Rule
and always have a -convex or, respectively, -coherent natural extension.
The role of these extensions is analogous to that of the natural extension for
coherent lower previsions. On the contrary, -convex and -coherent
previsions with either are convex or coherent themselves or have no
extension of the same type on large enough sets. Among the uncertainty concepts
that can be modelled by -convexity, we discuss generalizations of capacities
and niveloids to a conditional framework and show that the well-known risk
measure Value-at-Risk only guarantees to be centered -convex. In the final
part, we determine the rationality requirements of -convexity and
-coherence from a desirability perspective, emphasising how they weaken
those of (Williams) coherence.Comment: This is the authors' version of a work that was accepted for
publication in the International Journal of Approximate Reasoning, vol. 77,
October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003,
http://www.sciencedirect.com/science/article/pii/S0888613X1630079
Weak consistency for imprecise conditional previsions
In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of n-coherent and n-convex conditional previsions, at the varying of n. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) 2-convex or, if positive homogeneity and conjugacy is needed, 2-coherent lower previsions. Basic properties of these previsions are studied. In particular, centered 2-convex previsions satisfy the Generalized Bayes Rule and always have a 2-convex natural extension. We discuss then the rationality requirements of 2-convexity and 2-coherence from a desirability perspective. Among the uncertainty concepts that can be modelled by 2-convexity, we mention generalizations of capacities and niveloids to a conditional framework
The Goodman-Nguyen Relation within Imprecise Probability Theory
The Goodman-Nguyen relation is a partial order generalising the implication
(inclusion) relation to conditional events. As such, with precise probabilities
it both induces an agreeing probability ordering and is a key tool in a certain
common extension problem. Most previous work involving this relation is
concerned with either conditional event algebras or precise probabilities. We
investigate here its role within imprecise probability theory, first in the
framework of conditional events and then proposing a generalisation of the
Goodman-Nguyen relation to conditional gambles. It turns out that this relation
induces an agreeing ordering on coherent or C-convex conditional imprecise
previsions. In a standard inferential problem with conditional events, it lets
us determine the natural extension, as well as an upper extension. With
conditional gambles, it is useful in deriving a number of inferential
inequalities.Comment: Published version:
http://www.sciencedirect.com/science/article/pii/S0888613X1400101
Factorisation properties of the strong product
We investigate a number of factorisation conditions in the frame- work of sets of probability measures, or coherent lower previsions, with finite referential spaces. We show that the so-called strong product constitutes one way to combine a number of marginal coherent lower previsions into an independent joint lower prevision, and we prove that under some conditions it is the only independent product that satisfies the factorisation conditions
Updating beliefs with incomplete observations
Currently, there is renewed interest in the problem, raised by Shafer in
1985, of updating probabilities when observations are incomplete. This is a
fundamental problem in general, and of particular interest for Bayesian
networks. Recently, Grunwald and Halpern have shown that commonly used updating
strategies fail in this case, except under very special assumptions. In this
paper we propose a new method for updating probabilities with incomplete
observations. Our approach is deliberately conservative: we make no assumptions
about the so-called incompleteness mechanism that associates complete with
incomplete observations. We model our ignorance about this mechanism by a
vacuous lower prevision, a tool from the theory of imprecise probabilities, and
we use only coherence arguments to turn prior into posterior probabilities. In
general, this new approach to updating produces lower and upper posterior
probabilities and expectations, as well as partially determinate decisions.
This is a logical consequence of the existing ignorance about the
incompleteness mechanism. We apply the new approach to the problem of
classification of new evidence in probabilistic expert systems, where it leads
to a new, so-called conservative updating rule. In the special case of Bayesian
networks constructed using expert knowledge, we provide an exact algorithm for
classification based on our updating rule, which has linear-time complexity for
a class of networks wider than polytrees. This result is then extended to the
more general framework of credal networks, where computations are often much
harder than with Bayesian nets. Using an example, we show that our rule appears
to provide a solid basis for reliable updating with incomplete observations,
when no strong assumptions about the incompleteness mechanism are justified.Comment: Replaced with extended versio
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
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
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