Dear Colleagues,
In my 1990 book I defined a Bayesian network approximately as follows:
Definition of Markov Condition: Suppose we have a joint probability
distribution P of the random variables in some set V and a DAG G=(V,E). We
say that (G,P) satisfies the Markov condition if for each variable X in V,
{X} is conditionally independent of the set of all its nondescendants given
the set of all its parents.
Definition of Bayesian Network: Let P be a joint probability distribution
of the random variables in some set V, and G=(V,E) be a DAG. We call (G,P)
a Bayesian network if (G,P)satisfies the Markov condition.
The fact that the joint is the product of the conditionals is then an iff
theorem.
I used the same definition in my current book. However, a reviewer
commented that this was nonstandard and unintuitive. The reviewer suggested
I define it as a DAG along with specified conditional distributions (which
I realize is more often done). My definition would then be an iff theorem.
My reason for defining it the way I did is that I feel `causal' networks
exist in nature without anyone specifying conditional probability
distributions. We identify them by noting that the conditional
independencies exist, not by seeing if the joint is the product of the
conditionals. So to me the conditional independencies are the more basic
concept.
However, a researcher, with whom I discussed this, noted that telling a
person what numbers you plan to store at each node is not provable from my
definition, yet it should be part of the definition as Bayes Nets are not
only statistical objects, they are computational objects.
I am left undecided about which definition seems more appropriate. I would
appreciate comments from the general community.
Sincerely,
Rich Neapolitan
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