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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Enrique Miranda, Marco Zaffalon

# Coherence graphs

### Abstract

We consider the task of proving Walley's (joint or
strong) coherence of a number of probabilistic
assessments, when these assessments are represented as a collection
of conditional lower previsions. In order to maintain generality in
the analysis, we assume to be given nearly no information about the
numbers that make up the lower previsions in the collection. Under
this condition, we investigate the extent to which the above global
task can be decomposed into simpler and more local ones. This is
done by introducing a graphical representation of the conditional
lower previsions, that we call the coherence graph: we show
that the coherence graph allows one to isolate some subsets of the
collection whose coherence is sufficient for the coherence of all
the assessments. The situation is shown to be completely analogous
in the case of Walley's notion of weak coherence, for which
we prove in addition that the subsets found are optimal, in the
sense that they embody the maximal degree to which the task of
checking weak coherence can be decomposed. In doing all of this, we
obtain a number of related results: we give a new characterisation
of weak coherence; we characterise, by means of a special kind of
coherence graph, when the local notion of separate coherence
is sufficient for coherence; and we provide an envelope theorem for
collections of lower previsions whose graph is of the latter type.

** Keywords. ** Walley's coherence, weak coherence, coherent lower previsions, graphical models, coherence graph

** Paper Download **

The paper is availabe in the following formats:

** Authors addresses: **

Enrique Miranda

Dpto. de Informática, Estadística y Telemática

Univ. Rey Juan Carlos

C-Tulipán, s/n

Móstoles (Madrid)

SPAIN

Marco Zaffalon

Galleria 2

CH-6928 Manno

Switzerland

** E-mail addresses: **

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