Dear David,
an example of the collection of conditionals with no joint probability
measure can be obtained from an example of two compatible probability
measures P(x_1,x_2) and Q(x_2,x_3) [by compatibility is meant that
the marginals P(x_2) and Q(x_2) coincide] with no probability measure
R(x_1,x_2,x_3) having P and Q as marginals.
Indeed, then consider a DAG 1-->2-->3 and take the conditionals P(x_1),
P(x_2|x_1) and Q(x_3|x_2). No joint probability measure having these
conditionals exists then (otherwise it has P and Q as marginals).
An example of a pair of compatible probability measures P and Q of this
kind can be found e.g. as an Example B.1 in paper
A.P.Dawid, M.Studeny: Conditional products, an alternative approach
to conditional independence, in Artificial Intelligence and
Statistics 99 (D.Heckerman, J.Whittaker eds), Morgan Kaufmann,
pp. 32-40.
In fact, it is based on a construction of ugly measurable space and it
is an old counter-example mentioned in information theory
in 1960'ties. I myself gave the example in my Phd thesis in 1987 as well.
Nevertheless, one can hardly meet this nasty phenomenon in practice when
either discrete variables and continuous Gaussian variables are treated
(this includes the case of conditional Gaussian distributions). However,
it is important from a theoretical point of view, I guess. Simply, one can
always compute conditionals from a joint probability measure but not
conversely in general although this is possible in usual situations.
I agree with your opinion that Rich should ask students which definition
helps better to understand the concept.
Regards from
Milan
On Wed, 25 Jul 2001, David Poole wrote:
> 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
>
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