Rich,
One thing I really dislike about the "standard stat text" definition
of random variables and distributions is that the random variables
themselves change when the sample space is incrementally extended.
Suppose I have defined a belief network for inferring what disease a
patient has from a bunch of diseases, symptoms and background
variables. Now some doctor comes along and tells me he has been
seeing a brand-new skin rash no one has ever seen before in his
clinic. We keep an eye out and sure enough, this skin rash starts
cropping up all over the place. We then discover it is due to a
virus that used to infect only squirrels and is spread by fleas, but
has now mutated to infect the human population.
So we add a new disease to our list of diseases and a new symptom to
our list of symptoms, which means the BN now has two new variables it
didn't have before. We add some arcs connecting relevant background
information (such as region to the country and whether patient has
spent time outdoors) to these new symptoms. The rest of the BN stays
the same.
According to the standard statistics texts, I now have a new sample
space, which means all my random variables (including ones bearing no
relation to the new disease) are now different mathematical objects
from what they were before. "Mammogram," for example, used to be a
function from the old sample space (the cross-product of all the
state spaces of the previously modeled symptoms, background variables
and diseases) to the values "positive, negative, inconclusive." Now
it's a function from the new sample space (a cross-product with 2
additional dimensions for my two new random variables) to the same
outcome set.
I find it much less confusing to use the incremental specification as
the basic definition. Your defining things that way was one thing I
liked about your old text (which I used before it went out of print
and I couldn't get it any more). However, neither way of doing it is
"right." I tell students it can be done either way, because either
way is a valid way of specifying a joint probability distribution.
Kathy Laskey
