Dear David,
I think your feelings about `not having a belief network if you only have
the joint' lies at heart of individual differences on this matter. It seems
to the mathematician you most certainly do, and, in fact, that is part of
her problem with defining a Bayesian network in terms of conditionals.
However, to the `engineer' you don't have a Bayesian network unless it can
be constructed practically. I don't think there is any right or wrong here.
It is a matter of perspective. After all, even mathematicians sometimes
disagree on issues like this. E.g. some do not accept the axiom of choice.
I have received a lot of responses to my original query, but I do not know
how many went only to me. So I will report the result. It seems individuals
are divided about equally on this issue, with many being fairly certain
only one of the definitions is acceptable. It reminds me of what someone (I
think Ross Shachter) told me years ago, when I first became interested in
this field: "There is nothing researchers are more certain about than
uncertainty."
Sincerely,
Rich
p.s. In answer to your last recommendation, I agree. Indeed, I've been
teaching this material for the past 12 years, and I found students seem to
respond best when I define a Bayes net as a DAG plus a P that satisfies the
Markov condition with the DAG. Then I give an urn example of a P which does
do this with a DAG. Next I show P is the product of its conditional
distributions. Then I give the iff theorem. After this, I give the
practical applications. I state causal DAGs seems to exist in nature and
the actual relative frequency distribution of the variables seems to
satisfy the Markov condition with these DAGs. We can sometimes identify the
independencies and build the causal DAG, and we can estimate the
conditional distributions in the DAG. Although the resultant probability
distribution is only an estimate of the actual relative frequency
distribution that is `out there', due to the theorem, it still satisfies
the Markov condition with the DAG. So we end up with a Bayesian network
that represents our beliefs about the actual causal network in nature.
Maybe it is due to their training in classical statistics course, but, if I
take the other approach and just give them a DAG and conditionals to begin
with, they seem to think there is something hokey about the resultant
distribution, like it is just made up, and who cares if we have this
mathematical result that it satisfies the Markov condition with the DAG.
At 09:23 AM 7/25/01 -0700, David Poole wrote:
>
>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
>
>- --
>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
>
>
>
Rich Neapolitan
Computer Science Department
Northeastern Illinois University
5500 N. St. Louis
Chicago, Il 60625
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