Dear Rich,
in my view (I am a mathematician) your original definition of Bayesian
network is much more elegant than the other one which specifies
conditionals. So, I share the opinion of Marco Valtora.
My arguments in favour of the first definition are as follows.
The second approach does not seem suitable if one tries to go
behind discrete framework and to consider continuous Gaussian random
variables as some statisticians do. In fact, if one considers completely
general probablistic framework (random variables taking values in
general measurable spaces) then the concept of conditional probability
is quite complicated concept (from technical point of view) and the
described approaches are not equivalent in sense that a collection of
conditionals may exist for which no joint probability measure exists!
Your reviewer probably wanted to emphasize computational aspects.
Perhaps a good compromise is to enlarge your origninal definition
of Bayesian network by explicit specification of the sample space
X_i for every variable i. For example, (in discrete case) Bayesian
network could be introduced as a triple:
- a DAG over a set of variables N
- a set of possible values X_i for every variable i from N
- a probability distribution on the cartesian product of sets X_i
which satisfies ......
Then a method of storing a probability distribution of this kind
in memory of a computer which uses conditionals can be metioned.
(note that the conditionals are not unique in case that some configuration
have probability 0).
Regared from
Milan Studeny
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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