Take the standard math/CS approach: the thing we write down is
a piece of syntax; undoubtedly, we write down DAG (plus variable domains)
plus conditional distributions. The semantics maps the piece of syntax
to a joint probability distribution. One form of the semantics
(what AIMA calls the local semantics) *relies* on
stating that the DAG represents various
conditional independence assertions; so it is not as if we
are making conditional independence less basic. Without the
conditional independence assertions represented by the DAG part of the
syntax, no particular joint is represented by the numbers.
The other form of the semantics - what AIMA calls the global semantics -
simply says that the Bayes net (same piece of syntax)
represents the joint as the product of conditionals.
And these two forms of semantics are identical - one implies the other.
This way you avoid confusion about what the Bayes net *is* (it's a piece
of syntax); and you reconcile the two views of how to interpret that
syntax.
stuart
On Wed, 18 Jul 2001 profrich@megsinet.net wrote:
> 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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