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Having defined the Bayesian network, let's
look at some of the reasoning patterns

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that allow models to perform. So lets go
back to our good old student network

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with the following CPDs. We have already
seen those, so I'm not gonna dwell on it,

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and lets look at some of the probabilities
that one would get if you took the

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Bayesian network and produced the joint
distribution using the chain rule for

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a Bayesian network and now we've computed and saved
the values of different marginal

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probabilities. So, for example now we are
asking what is the probability of getting

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a strong letter, and we're not going to go
through the calculation, because it's

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going to be tedious to sum up all these
numbers, and I can just tell you that the

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probability of the of l1 is 0.5. But we
can do more interesting queries, so we can

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now condition on one variable, remember we
talked about conditioning and probability

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distribution, and ask how that
changes this probability. So, for example,

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if we're going to condition on low
intelligence, we're going to use red to

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denote the "false" value.

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And it's going to affect the good letter's
probability, it turns out the probability, not

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surprisingly, goes down. It goes down to
0.39, because if the intelligence goes down,

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the probability of getting a good grade
goes down and so does the probability of

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getting a strong letter. So, this is an example
of causal reasoning because intuitively,

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the reasoning goes in the causal direction, from
top to bottom. We could also make things

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more interesting. So we can ask what
happens if we make the difficulty of the

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course low. And in this case, we have the
probability of l1 given i0 and d0,

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and what you expect the
probability to do, well, if it's an easy

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course, one would expect the grade to go
up. And sure enough, the probability goes

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back up, and we're back to 50/50, more or
less. Okay, so this is another example of

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causal reasoning, in this case,
with a little bit more evidence. You can

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also do evidential reasoning, evidential
goes from the bottom up. So we can, in this

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case, condition on the grade and ask what
happens to the probability of variables

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that are parents or in general,
ancestors of the grade. So let's just

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imagine, let's suppose if a student takes
the class and he gets a C, initially the

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probability that the class was difficult is
0.4 and the probability if the student was

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intelligent is 0.3 but now with this
additional evidence, again this is not

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surprising, the probability that the student is
intelligent goes down by a fair amount.

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The other, alternative, hypothesis that the
class is difficult, also the probability of

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that goes up as well. So. Now,
however, there is an interesting type of

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reasoning that is not quite as standard,
and that is reasoning that is called

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intercausal because effectively, it is
flow of information between two causes of

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a single effect. So, remember, we have
the -- we're going to continue with the

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scenario before where our poor student
gets a C. It's now going to tell you: wait

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a minute, this class really is difficult.
So I'm going to condition on d1. And

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notice that the probability of the student, his
intelligence has gone up. It went up from

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0.08 to 0.11. So that's not a huge
increase. And as you'll see when you play

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around with Bayesian Networks, often the
changes in probability are somewhat

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subtle. And the reason is that you have to --
I mean, even in a hard class, if you go back

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and look at the CPD, it's kinda hard to
get a C according to this model. Which is

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that the student gets a B. And so now, we
have that the probability of high intelligence

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still goes down, it goes down from 0.3 to
0.175, but now, if I tell you the class is

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hard, the probability goes up. In fact, it
goes up even higher than this, okay? So

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this is an illustration where this 
intercausal reasoning can actually

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make a fairly significant difference in
the probabilities. So intercausal

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reasoning is a little hard to understand,
I mean, it seems a little bit mysterious,

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because, after all,
you look at these, you look at difficulties, you

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look at intelligence, there is no edge
between them. How would one cause affect another?

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So let's drill down
into a concrete scenario, which is this one,

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and just, to sort of really understand the
mechanism. So this is the most, sort of

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purest form, of intercausal reasoning.
Here we have two random variables, X1 and

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X2. We're going to assume that they're
distributed uniformly, so each of them is

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one with probability of 50% and
zero with probability of 50%, and we have

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one effect, one joined effect, which is
simply the deterministic or of those two

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parents. And in general, when we have the
deterministic variable, we're going to

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denote it with these double
lines. So, in this case there's only four

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assignments that have non-zero
probability, because the value of Y is

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completely determined by the values of
X1 and X2. And so, we have these

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four distributions over here, and now I'm
going to condition on the evidence Y=1.

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Now, let's look at what happened. Before I
conditioned on this evidence, [X1 and X2]

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were independent of each other, right? I
mean, look at this, they're independent of

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each other. What happens when I condition
on Y=1? Well, we talked about

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conditioning. This one goes away, and we
have 0.33, 0.33, 0.33,  or rather one-third,

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one-third, one-third. Okay, in this
probability distribution X1 and X2 are no

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longer independent of each other. Okay?
Why is that? Because if I now condition on,

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say X1=0,  then -- okay, so,
actually before we do that, so the, in

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this probabilty distribution the
probability of X1=1 is equal to

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2/3, and the probabilty of X2=1
is also equal to 2/3. And now if I

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condition on X1=1... so now we're
going to condition on condition X1=1,

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so that means we're going to remove this
line and then, all of a sudden, the probability

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of X2=1 given X1=1 is back to
being 50%. So it was 50% before. It

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went up to two-thirds, and then, if we
condition on X1 = 1, it goes back to

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50%. And the reason for this is the
following, if you think about it

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intuitively: If I know that Y is equal to
one there's two possible things that could

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have made Y equal to one. Either X1 was
one or X2 was one. If I've told you that

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X1=1, I've completely explained away
the evidence that Y = 1, I've given you

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a complete explanation of what happened
and so now it's going back to being the

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way it was before because there is no
longer anything to suggest that it should

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be anything other than 50/50. So, this
particular type of intercausal reasoning, because

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it's so common, it's called "explaining
away". And it's where one cause

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explains away the reason that made me
suspect a different cause. And if you

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think about it, it's something that people
do all the time when they are reasoning about,

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for example, on a medical setting, you,
you're very sick, you think, you're very

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worried, you don't know if
you have the swine flu. You go to the

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doctor, the doctor says "oh, don't worry
it's just a common cold". You don't know

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that you don't have the swine flu, but because
you've explained away your symptoms you're

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not worried as much any more. Finally,
let's look, lets go back to our example and

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look at an interesting reasoning pattern
that is not -- that involves even longer sort

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of paths in the graph, so let's imagine
that we have this student the student got

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a C, but now we have this additional piece
of information that the student actually

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aced the SAT, so hopefully what happens
there. Remember that when we just had the

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evidence regarding the grade, we had the
probability of the student being

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intelligent was only 0.08. But now we have
this additional conflicting piece of

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evidence. And all of a sudden, the
probability went up very dramatically to

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0.58. Okay. What do you think is going to
happen to difficulty?

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So now, it's explaining away in action, going in a
different direction, right? Because if

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it's not the fact that the student -- I mean,
if the student didn't get a C because he

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wasn't very bright, probably the reason is
that the class is very difficult. And so

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that probability goes up and so we have
effectively, and we're gonna talk about

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this, an inference that flows like that.