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, x_{1} and x_{2}, with no arrow between
them (Figure 1).
x_{1} is associated with a probability distribution p_{1}(x_{2}) and
x_{2} is associated with a probability distribution p_{2}(x_{2}).
A global density ratio neighborhood is constructed:

(1/k) p_{1}(x_{1}) p(x_{2}) lealphar(x_{1}, x_{2}) lek p_{1}(x_{1}) p(x_{2}).

© Fabio Cozman[Send Mail?]

Thu Jan 23 15:54:13 EST 1997