There is a fundamental difference here.
Suppose you have two variables, say A and B for which you have the joint
distribution P(A,B). I would claim that you don't have a belief network
(Bayes net) unless you have P(A) and P(B|A) or P(B) and P(A|B).
Why should we care? If A and B are such that we can have the cumulative
probability distribution (e.g., thet are on subsets of the reals) then
we can easily do stochastic simulation (e.g., logic sampling) if we have
a belief network as above. (We can generate a random sample by
generating two random numbers in [0,1]). If we just have P(A,B) then is
difficult to do stochastic simulation and we have to revert to methods
such as MCMC. Such distributions where we can't easily specify them as
a belief network arise very naturally when we are learning the structure
of belief networks.
Milan Studeny wrote:
> 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!
Milan, Do you have an example or a reference for such a collection of
conditionals?
So Rich, to answer your question, you should think of whether all joint
distributions are belief networks or just those ones for which you have
the conditionals. As Milan says, sometimes the conditionals are hard to
come by (in practice as well as in theory).
But I think we are exactly the wrong people to ask about what is a
natural definition of a belief network! You should be asking the people
who don't know about them whether your definition makes sense or helps
them understand the concept. (Of course you don't want to define
something that doesn't coincide with the normal definitions).
Good luck with your book.
David
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David Poole, Office: +1 (604) 822-6254
Department of Computer Science, poole@cs.ubc.ca
University of British Columbia, http://www.cs.ubc.ca/spider/poole
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