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# Independence relations

A Bayesian network exhibits several independence relations in a joint distribution. In general, neighborhoods of a Bayesian network do not preserve all the decompositions in the original joint distribution [Berger & Moreno1994, Walley1991]. This fact has led to considerable discussion in the literature [Chrisman1996a, Seidenfeld & Wasserman1993]. Here it should not disturb us, as we are interested in creating neighborhoods of probabilistic models: it is not surpr is in g that a neighborhood of a model should contain some distributions which do not obey exactly the same properties of the model.

Figure 1: Bayesian network with two independence variables

To illustrate this point, consider a network with two variables, x1 and x2, with no arrow between them (Figure 1). x1 is associated with a probability distribution p1(x2) and x2 is associated with a probability distribution p2(x2). A global density ratio neighborhood is constructed:

(1/k) p1(x1) p(x2) lealphar(x1, x2) lek p1(x1) p(x2).

This neighborhood includes many models r(x1, x2) where x1 and x2 are related and dependent.