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There is no unique way to define independence relations with credal sets;
in the most in-depth study of this matter, Campos and Moral have reviewed
five different possible types of independence [10].
The results presented in this paper adopt Walley's definition of
independence [28]. Walley's original definition was stated in terms
of lower expectations; to develop a theory of convex sets of distributions,
it is important to recast Walley's definition using credal sets as follows.
Consider sets of variables ,
and
and the credal sets
,
,
,
and
.
Note that distributions
in
and
are
defined over the same algebra of events
once
and
are fixed; likewise, distributions
in
and
are defined over the same algebra of events once
and
are fixed.
Definition 1
Variables
are
irrelevant to
given
if
is equal to
regardless of the value of
,
.
Variables
and
are
independent given
if
is
irrelevant to
given
and
is irrelevant
to
given
.
If
is empty, suppress the
``given
'' from this definition.
This concept of independence does not imply that joint credal sets
contain only joint distributions with independent marginals, nor does
it imply uniqueness for the joint credal set [28, Chapter 9].
Next: LOCALLY DEFINED QUASI-BAYESIAN NETWORKS
Up: QUASI-BAYESIAN THEORY
Previous: Conditionalization
Fabio Gagliardi Cozman
1998-07-03