Having defined the Bayesian network, let's
look at some of the reasoning patterns
that allow models to perform. So lets go
back to our good old student network
with the following CPDs. We have already
seen those, so I'm not gonna dwell on it,
and lets look at some of the probabilities
that one would get if you took the
Bayesian network and produced the joint
distribution using the chain rule for
a Bayesian network and now we've computed and saved
the values of different marginal
probabilities. So, for example now we are
asking what is the probability of getting
a strong letter, and we're not going to go
through the calculation, because it's
going to be tedious to sum up all these
numbers, and I can just tell you that the
probability of the of l1 is 0.5. But we
can do more interesting queries, so we can
now condition on one variable, remember we
talked about conditioning and probability
distribution, and ask how that
changes this probability. So, for example,
if we're going to condition on low
intelligence, we're going to use red to
denote the "false" value.
And it's going to affect the good letter's
probability, it turns out the probability, not
surprisingly, goes down. It goes down to
0.39, because if the intelligence goes down,
the probability of getting a good grade
goes down and so does the probability of
getting a strong letter. So, this is an example
of causal reasoning because intuitively,
the reasoning goes in the causal direction, from
top to bottom. We could also make things
more interesting. So we can ask what
happens if we make the difficulty of the
course low. And in this case, we have the
probability of l1 given i0 and d0,
and what you expect the
probability to do, well, if it's an easy
course, one would expect the grade to go
up. And sure enough, the probability goes
back up, and we're back to 50/50, more or
less. Okay, so this is another example of
causal reasoning, in this case,
with a little bit more evidence. You can
also do evidential reasoning, evidential
goes from the bottom up. So we can, in this
case, condition on the grade and ask what
happens to the probability of variables
that are parents or in general,
ancestors of the grade. So let's just
imagine, let's suppose if a student takes
the class and he gets a C, initially the
probability that the class was difficult is
0.4 and the probability if the student was
intelligent is 0.3 but now with this
additional evidence, again this is not
surprising, the probability that the student is
intelligent goes down by a fair amount.
The other, alternative, hypothesis that the
class is difficult, also the probability of
that goes up as well. So. Now,
however, there is an interesting type of
reasoning that is not quite as standard,
and that is reasoning that is called
intercausal because effectively, it is
flow of information between two causes of
a single effect. So, remember, we have
the -- we're going to continue with the
scenario before where our poor student
gets a C. It's now going to tell you: wait
a minute, this class really is difficult.
So I'm going to condition on d1. And
notice that the probability of the student, his
intelligence has gone up. It went up from
0.08 to 0.11. So that's not a huge
increase. And as you'll see when you play
around with Bayesian Networks, often the
changes in probability are somewhat
subtle. And the reason is that you have to --
I mean, even in a hard class, if you go back
and look at the CPD, it's kinda hard to
get a C according to this model. Which is
that the student gets a B. And so now, we
have that the probability of high intelligence
still goes down, it goes down from 0.3 to
0.175, but now, if I tell you the class is
hard, the probability goes up. In fact, it
goes up even higher than this, okay? So
this is an illustration where this 
intercausal reasoning can actually
make a fairly significant difference in
the probabilities. So intercausal
reasoning is a little hard to understand,
I mean, it seems a little bit mysterious,
because, after all,
you look at these, you look at difficulties, you
look at intelligence, there is no edge
between them. How would one cause affect another?
So let's drill down
into a concrete scenario, which is this one,
and just, to sort of really understand the
mechanism. So this is the most, sort of
purest form, of intercausal reasoning.
Here we have two random variables, X1 and
X2. We're going to assume that they're
distributed uniformly, so each of them is
one with probability of 50% and
zero with probability of 50%, and we have
one effect, one joined effect, which is
simply the deterministic or of those two
parents. And in general, when we have the
deterministic variable, we're going to
denote it with these double
lines. So, in this case there's only four
assignments that have non-zero
probability, because the value of Y is
completely determined by the values of
X1 and X2. And so, we have these
four distributions over here, and now I'm
going to condition on the evidence Y=1.
Now, let's look at what happened. Before I
conditioned on this evidence, [X1 and X2]
were independent of each other, right? I
mean, look at this, they're independent of
each other. What happens when I condition
on Y=1? Well, we talked about
conditioning. This one goes away, and we
have 0.33, 0.33, 0.33,  or rather one-third,
one-third, one-third. Okay, in this
probability distribution X1 and X2 are no
longer independent of each other. Okay?
Why is that? Because if I now condition on,
say X1=0,  then -- okay, so,
actually before we do that, so the, in
this probabilty distribution the
probability of X1=1 is equal to
2/3, and the probabilty of X2=1
is also equal to 2/3. And now if I
condition on X1=1... so now we're
going to condition on condition X1=1,
so that means we're going to remove this
line and then, all of a sudden, the probability
of X2=1 given X1=1 is back to
being 50%. So it was 50% before. It
went up to two-thirds, and then, if we
condition on X1 = 1, it goes back to
50%. And the reason for this is the
following, if you think about it
intuitively: If I know that Y is equal to
one there's two possible things that could
have made Y equal to one. Either X1 was
one or X2 was one. If I've told you that
X1=1, I've completely explained away
the evidence that Y = 1, I've given you
a complete explanation of what happened
and so now it's going back to being the
way it was before because there is no
longer anything to suggest that it should
be anything other than 50/50. So, this
particular type of intercausal reasoning, because
it's so common, it's called "explaining
away". And it's where one cause
explains away the reason that made me
suspect a different cause. And if you
think about it, it's something that people
do all the time when they are reasoning about,
for example, on a medical setting, you,
you're very sick, you think, you're very
worried, you don't know if
you have the swine flu. You go to the
doctor, the doctor says "oh, don't worry
it's just a common cold". You don't know
that you don't have the swine flu, but because
you've explained away your symptoms you're
not worried as much any more. Finally,
let's look, lets go back to our example and
look at an interesting reasoning pattern
that is not -- that involves even longer sort
of paths in the graph, so let's imagine
that we have this student the student got
a C, but now we have this additional piece
of information that the student actually
aced the SAT, so hopefully what happens
there. Remember that when we just had the
evidence regarding the grade, we had the
probability of the student being
intelligent was only 0.08. But now we have
this additional conflicting piece of
evidence. And all of a sudden, the
probability went up very dramatically to
0.58. Okay. What do you think is going to
happen to difficulty?
So now, it's explaining away in action, going in a
different direction, right? Because if
it's not the fact that the student -- I mean,
if the student didn't get a C because he
wasn't very bright, probably the reason is
that the class is very difficult. And so
that probability goes up and so we have
effectively, and we're gonna talk about
this, an inference that flows like that.