This is an attempt to answer a question of Robert Dodier whether
there exists a simplified version of a couter-example I have mentioned
in the discussion. Please skip it if you are not interersted in it.
On Thu, 26 Jul 2001, Robert Dodier wrote:
> Hello Milan,
>
> I am following the discussion of the definition of Bayesian network.
> I have looked at your paper (Dawid and Studeny) in which you give
> an example of conditional distributions for which the joint as a
> product of the conditionals does not exist. However, I am unable
> to fully understand the example (Appendix B.2). Can you post to
> the UAI email list a simplified discussion? I wonder if there is
> not a simple, concrete example which illustrates the concept.
>
> The discussion is very interesting, and I appreciate very
> much your contribution.
>
> Regards,
> Robert Dodier
>
Dear Robert,
thank you for your interrest. The message of my contribution to the
discussion was that in completely general probabilistic case (when one
considers general measurable spaces as possible sample spaces) and which
is fortunately far behind discrete case treated in practice in
probabilistic reasoning a strange phenomenon may occur. This phenomenon
cannot occur in discrete case (when every variable can take only finitely
many values and one has finitely many variables) or in Gaussian case which
is treated inmathematical statistics. I think it is quite important to
mention this for practitioners!
What may happen in general case is that a collection of conditionals may
exist such that no joint probability meaasure having them as marginals
exists. The constructiion could be based on an example of two compatible
probability measures which do not have a joint measure having them as
marginals. Note that another example of the collection of conditionals
could be based on an example of a joint 3-dimensional measure P over 1,2,3
such that there is no probability measure R sharing marginals of P for
1,2 and 2,3 in which 1 is conditionally independent of 3 given 2.
However, to be honest both examples mentioned above go deeply in
technical foundations of measure theory and use specific construction of
'ugly' probability measures from Halmos's book 'Measure Theory'. This
construction uses axiom of choice. Therefore the examples from (Dawid
Studeny) cannot be fully understood without this background. Well, I also
think that one can hardly find a 'simplified' example without axiom of
choice.
The reason why I emphasized those counterexamples was that an analogous
situation may occur in other uncetainty calculi - e.g. possibility theory
- - and researchers in UAI community should be aware of this danger when
they 'generalize' too much.
Perhaps I can explain in vague words the main reason why one cannot
define the joint of conditionals as a product (which is one yours
formulations, Robert). The reason is that the product of conditionals
which can be easily introduced in discrete case appears to be a technical
complicacy in general. I have already mentioned earlier that the concept
of condtional (= cconditional probability) is quite complicated
mathematical concept in general. In fact, one meets techincalities already
in case of discrete variables, I guess.
Measure theory gives a solution of the problem of defining the product
of conditionals: first, one has to define the joint maesure on so-called
'measurable rectangles' and then to extend this set funtion to all
measurable sets. This extension, which is the resulting joint probability
merasure, exists if and only if the 'extended' set function on measurable
rectangles id sigma-additive. The point ofabove mentioend examples is that
the set function on measurable rectangles determined by the given
conditionals (by the requirement that is a product of conditionals)
NEED NOT be sigma-additive! In that case no joint probability meausre
of the 'product form' exist althoug one can find many finitely-additive
non-negative 'joint' measure which extend the above mentioned set function
on measurable rectangles.
I hope that these remarks describe the essence of those counterexamples
enough.
Milan
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