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

Â patterns of work allows us to perform. So let's go back to our good old student

Â network with the following CPDs. We've already seen those.

Â So I'm not going to dwell on it. And let's look at some of the

Â probabilities that one would get if you took this busy network, produced the

Â joined distribution using the chain rule for busy network.

Â And now compute it say the values of different marginal probabilities.

Â So for example now we're asking what is the probability of getting a strong

Â letter, and we're not going to go through the calculation, because wanting to be

Â tedious to sum up all these numbers and I can just tell you that the probability of

Â the the of L1, is 0.5 but we can do more interesting queries.

Â So we can now condition on one variable. Remember we talked about conditioning of

Â probability distribution. And and ask how that changes this

Â probability. So for example, if we're going to

Â condition on low intelligence, we're going to use red to denote the false

Â value. And it's going to point to turn out that

Â the probability not surprisingly goes down.

Â It goes down to 0.39 because if the intelligence goes down, the probability

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

Â letter. So this is an example of causal

Â reasoning, because because, intuitively the reasoning goes, in the causal

Â direction from top to bottom. We could also make things more

Â interesting. So we can ask what happens if we make the

Â difficulty of the course low and in this case, we have the probability of L1,

Â given i0 and b0. And what you expect the probabilities to

Â do, well, if it's an easy course, one would expect the grade to go up.

Â And sure enough the probability goes back up and we're back to 50/50, more or less.

Â Okay, so this is another example, of [UNKNOWN] in this case with a little

Â more, evidence. You can also do evedentual reasoning.

Â Evedentual goes from the bottom up. So we can in this case condition on the

Â grade and ask what happens to the probability of, of variables that are

Â parents or, or general ancestors of the grade.

Â So does it matter that this poor student takes the class and he gets a C.

Â Initially the probability that the class was difficult is 0.4 and the probability

Â that the student was intelligent is 0.3. But now with this additional evidence,

Â again this is not surprising, the probability that the, that the student is

Â intelligent goes down a fair amount but the other alternative hypothesis, that

Â the class is difficult also the probability of that goes up as well.

Â Hm. Now however there is an interesting type

Â or reasoning that is not quite as standard.

Â And that is reasoning that is called inter-causal because effectively it's

Â flow of information between two causes of a. of a single effect.

Â So remember we had the we're going to continue with the scenario before where

Â our poor student gets a C but now I'm going to tell you, wait a minute.

Â This class really is difficult so I'm going to condition on on v1 and notice

Â that the propability that the student his intelligence has gone up, it went up from

Â 0.08 to 0.11 so that's not a huge increse and as you'll, see when you play around

Â with Bayesian networks, that often the changes in probability are somewhat

Â subtle. and the reason is that, you have to, I

Â mean, even in a hard class if you go back and look at the CPD it's kind of hard to

Â get a C, according to this model. which is that the students get a B.

Â and so now, have that the probability of high intelligence still goes down, it

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

Â probability goes up, in fact it goes up even higher than this, okay?

Â So, this is an illustration where this, where this intercausal reasoning can

Â actually make a fairly significant difference in the probabilities.

Â So intercausal reasoning is a little hard to understand, I mean she's a little bit

Â mysterious because after all, these are, I mean look at these you look at

Â difficulty you look at intelligence there's no edge between them how, how

Â would how would one cause affect another. So let's drill down into a concrete

Â scenario which is this one and just to sort of really understand the mechanism.

Â So, this is the most sort of purest form of intercausal reasoning.

Â Here we have two random variables x and, x1 and x2.

Â We're going to assume that they're distributed uniformally so each of them

Â is one with probability 50% and zero probability 50%.

Â And we have one effect one joined effect which is simply the deterministic oar of

Â those two parents. And in general we have the terministic

Â variable we're going to denote with these with these double lines.

Â So, in this case, there's only four assignments that have nonzero

Â probability, because, the value of Y is completely determined from, by the values

Â of X1 and X2. And so we have we have these four

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

Â Now let's look at what happened. Before I conditioned on this evidence,

Â the X2 were independent of each other, right?

Â I mean, look at this. They're independent of each other.

Â What happens when I condition on y = 1? Well we talked about conditioning.

Â This one goes away, and we have 0.33, 0.33.

Â 0.33 or rather one third, one third. Okay.

Â In this probability distribution x1 and x2 are no longer independent of each

Â other. Okay.

Â Why is that? Because if I now condition on say x1

Â equals 0, then okay, so actually before we do that,

Â so that, in, in this probability distribution the probability of x1 equals

Â 1 is equal to two-thirds and the probability of x, two equals one.

Â 6:55

Is also equal to two thirds. I think,

Â and now if I condition on x1 = 1. So now, we're going to condition on,

Â conditions x11. = 1.

Â So that means we're going to remove this line.

Â And all of a sudden, the probability of x21 = 1 given x11 = 1 is back to being

Â 50%.. So with 60% of 4, it went up to

Â two-thirds and then if we condition on x11, = 1, it goes back to 50%..

Â And the reason for this is the following, think about it intuitively.

Â If I know the y1 = 1 there's two possible things that could have made y1, = 1.

Â Either x1 was 1 or x2 was 1. If I've told you that X1 was one, I've

Â completely explained away the evidence that Y equals one.

Â I've given you a complete explanation of what happened, and so now I just want to

Â go back the way it was before, because there is no longer anything to suggest

Â that that it should be, anything other than 50/50.

Â So this particular type of [INAUDIBLE] because it's so common.

Â It's called explaining away. And it's when one cause explains a way

Â reasons that made me suspect a different cause.

Â And if you think about it it's something that people do all the time, when they're

Â reasoning about for example in the medical setting you, you're very sick,

Â you think you're very worried you have you don't know if you have the swine flu,

Â you go to the doctor the doctor says don't worry it's just the common cold.

Â You don't know if, that you don't have the swine flu, but because you have

Â explained away your symptoms you're not worried as much anymore.

Â Okay. Finally.

Â Lets look, lets go back to our example, and look at an interesting reasoning

Â pattern that is not that, that involves even longer sort of paths in the graph.

Â So lets imagine that we have the student, the student got a C.

Â But now we have this additional piece of information that the student actually

Â aced the SAT. So hopefully what happens there.

Â Remember that when we just had the evidence regarding the grade we had the,

Â the probability that the students being intelligence was only 0.08.

Â But now we have this additional conflicting piece of evidence, and all of

Â a sudden the probability went up very dramatically to 0.58, okay?

Â What do you think is going to happen to difficulty?

Â So, now it's explaining a way in action going in a different direction right?

Â Because if it's not the fact that student, I mean if the student didn't get

Â a C because he wasn't very bright, probably, the reason is that the class is

Â very difficult, and so that probability goes up.

Â And so we have effectively and we're going to talk about this an inference

Â that folds like that.

Â