Very human. So again your not forced to use it and
sometimes there is a good reasons not to do it but.
It's generally a good tip to follow. So how does one actually construct a
graphical model? Do we have in our minds some monolithic P
of some set of variables, X1 up to XN and we just need to figure out how to encode
that using a graph? Well maybe implicitly, but certainly not
in any explicit form. The way in which one typically constructs
a graphical model in practice is by having some variable or sometimes set of
variables that we wish to reason about. So, for example, we might care about the
variable cancer or maybe even lung cancer.
Well, what influences, whether we have cancer.
whether somebody is going to get lung cancer.
Well if we go an ask a doctor. What is the probability for someone to
get lung cancer? The doctor is going to say, well.
You know, that depends. And you might say, what does it depend
on? An the doctor will say, well.
Whether they smoke for example. At which point.
You're likely to add the variable smoking as a parent to the lung cancer variable.
The doctor might say well but that's not the only thing, it might the probability
of cancer also depends for example on the kind of work that you do because some
kinds of work involve more dust particles getting into your lungs and so again
here's another variable which you would add as a parent.
And I even go and ask either there a doctor or an expert in a different domain
what is the probability that somebody smokes?
And if they think about it they're likely to say that depends, and what does it
depend on? Well maybe their age, gender,
maybe their, the country that they live in because certain different countries
have different smoking frequencies. And so once again, we're going to extend the
conversation backward to include more variables up to the point that we can
stop, because if you now ask for example, what is the probability of gender being
male versus female, well anybody can answer that one.
And at that point one can stop because there's no way to extend the conversation
backward. Is that enough?
Usually not because we also need to consider for example, factors that might
help us might indicate to us whether somebody's going to have can, somebody
has cancer or not. And so we might go and ask the doctor
what are some pieces of evidence that might be indicative here, and we would,
the doctor would tell us for example, coughing or maybe bloody sputum and
various other things that would be potential indicators.
And at that point, one would say, well, okay.
What is the probability of coughing given lung cancer?
And again, one would now extend the conversation backward to say.
Well, other things may cause coughing. For example, having allergies.
And so once again we would, go from here and extend backward, to construct a
graphical model that captured, all the relavent factors for answering queries
that we hear about. So, that's the structure of a graphical
model now let's talk a little bit about parameters, the values of these
parameters and what make a difference here, so here are certain things that
really do make a difference, to parameters, zeros.
Make big difference. And when we talked about diagnosis we saw
that many of the mistakes that were made in early medical expert systems were
derived from the fact that people gave zeros to things that was unlikely.
But not actually impossible. And so zeros are something to be very,
very careful about. Because you should only use something,
you should only give probability zero to something that is, impossible perhaps
because it's definitional. Otherwise, things really shouldn't have
probability zero. Other things that make a difference are a
sort of weaker versions. So for example, orders of, order of
magnitude differences, the difference between a probability of one over ten
versus one over 100 that makes a difference.
It makes a much bigger, whereas small differences like 0.54 versus 0.57 are
unlikely to make a difference to most queries.
Finally it's turned out that relative values between conditional probabilities
make a much bigger difference to the answer than the absolute probabilities.
That is, the, comparing different entries in the same CPD, relative to each other,
is a very useful way of of evaluating the graphical model and seeing whether the
value. Use that you use for those relative
ratios really make sense. Finally,
Conditional probability tables are actually quite rare acceptance small
applications. In most cases one would use structured
CPDs of the forms that we've discussed as well as the variety of other forms.
So let's talk a little bit about structured CPDs because those are
actually quite important. and we can break up of the.
The types of CPD's that we've talked about along two axes: one is whether
they're intended to deal primarily discreet or with continuous variables.
And on the other side is whether they type of structure that they encode is
context specific, where a variable might make a difference in some circumstances
and not in others, versus aggregating. Of multiple weak influences.
And so let's give off an example of each of these categories.
So for discrete and context specific, we had three cpd's as an example.
For discrete and aggregating we had sigmoid.
CPD's as well as noisy or, where noisy max or any one of those, that family.
For continuous CPD's we didn't actually talk about context specific,
representations, but one can take the, continues version of tree CPD called a
regression tree. Where one breaks up the context based on
some threshold on the continuous variables.
