Having defined the Bayesian network, let's look at some of the reasoning
patterns of work allows us to perform. So let's go back to our good old student
network with the following CPDs. We've already seen those.
So I'm not going to dwell on it. And let's look at some of the
probabilities that one would get if you took this busy network, produced the
joined distribution using the chain rule for busy network.
And now compute it say the values of different marginal probabilities.
So for example now we're asking what is the probability of getting a strong
letter, and we're not going to go through the calculation, because wanting to be
tedious to sum up all these numbers and I can just tell you that the probability of
the 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 of
probability distribution. And 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 point to turn out that
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 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 b0. And what you expect the probabilities 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 [UNKNOWN] in this case with a little
more, evidence. You can also do evedentual reasoning.
Evedentual goes from the bottom up. So we can in this case condition on the
grade and ask what happens to the probability of, of variables that are
parents or, or general ancestors of the grade.
So does it matter that this poor student takes the class and he gets a C.
Initially the probability that the class was difficult is 0.4 and the probability
that the student was intelligent is 0.3. But now with this additional evidence,
again this is not surprising, the probability that the, that the student is
intelligent goes down a fair amount but the other alternative hypothesis, that
the class is difficult also the probability of that goes up as well.
Hm. Now however there is an interesting type
or reasoning that is not quite as standard.
And that is reasoning that is called inter-causal because effectively it's
flow of information between two causes of a. of a single effect.
So remember we had the we're going to continue with the scenario before where
our poor student gets a C but now I'm going to tell you, wait a minute.
This class really is difficult so I'm going to condition on on v1 and notice
that the propability that the student his intelligence has gone up, it went up from
0.08 to 0.11 so that's not a huge increse and as you'll, see when you play around
with Bayesian networks, that 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 kind of hard to
get a C, according to this model. which is that the students get a B.
and so now, 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, 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 she's a little bit
mysterious because after all, these are, I mean look at these you look at
difficulty you look at intelligence there's no edge between them how, how
would 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 x and, x1 and x2.
We're going to assume that they're distributed uniformally so each of them
is one with probability 50% and zero probability 50%.
And we have one effect one joined effect which is simply the deterministic oar of
those two parents. And in general we have the terministic
variable we're going to denote with these with these double lines.
So, in this case, there's only four assignments that have nonzero
probability, because, the value of Y is completely determined from, by the values
of X1 and X2. And so we have 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,
the 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. 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
equals 0, then okay, so actually before we do that,
so that, in, in this probability distribution the probability of x1 equals
1 is equal to two-thirds and the probability of x, two equals one.