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So we've seen reasoning patterns where intuitively,

at least that's how we argued, probabilistic influence kind of starts in

one node and flows through the graph to another node.

Now it might seem like, you know, a bunch of hand waving but it turns out this is

actually, exactly what goes on in a Bayesian network.

So what we're going to do now, we're going to make this argument much more

rigorous by trying to understand exactly when one variable X can influence another

variable Y. And we're going to start with a case

where there's no evidence going on and we're just asking can variable X

influence Y and lets look at a few simple cases.

So first, if X and Y are connected, so X say is a parent of Y, then yup,

pretty much. It's pretty clear cause this X can

influence Y. If Y is a child of X, we already talked

about evidential reasoning, we also saw, this case X can influence Y in the sense

that observing X can change my probability distribution of Y, that's

what I mean by influence. Influence means [INAUDIBLE] in Y or about Y.

[SOUND] More interesting are the cases where we have indirect influence between

X and Y. So let's consider a case where there's an

intervening variable W and let's think about can X influence Y via W.

And the first case is a causal chain, for example, such as, for example, one

going from difficulty to letter via grade.

And we've already seen that in this case, X can influence Y via W in for, in, in

examples that we saw. This is exactly the same idea, except

that we're going evidential and as we'll see in general probabilistic influence is

symmetrical, that is if X can influence Y, Y can

influence X. And so that's

so here also, we have probabilistic influence line.

Okay? The third, is a structure that looks like

this, so we have a common cause, w, that has

two effects X and Y. And, again, it seems to make sense that

if we observe the value of the SAT, then that changes my beliefs in the student's

intelligence and subsequently my probability distribution over their

grade. 'Kay? So the last and most interesting

one is this case, which is the case of two causes that have a joint effect.

This case is also called a V-structure, for obvious reasons,

because it's shaped like a V. [INAUDIBLE] remember, I haven't given you

any information. The question is, if I tell you that a

student took a class and the class is difficult,

does that tell me anything about the student's intelligence?

And the answer is no. And so this is, in this case, the one and

only exception in this particular case to allow us to define this notion of active

trail in the context of no evidence. So a trail in general, is a sequence of

nodes that are connected to each other by single edges in the graph.

So X1 up to actually we should make this Xk, so as not to confuse with a set of

variables, and the fact that these edges are undirected means that they can go in

either direction. So, I'm not stipulating that it goes up

or down. So, basically, we saw that influence can

flow from one variable to another variable in the graph and what this

definition basically says is that the is that the this influence can continue to

flow. So if it flows from one variable to the

next, to the next, to the next, to the next, that still defines an active trail.

The only thing that blocks an active trail is the V-structure, because that is

the one case where we have that no influence flows as in the example that we

showed before. So this is a block in the trail.

Now lets look at a more interesting case. Now, we have some set of observations,

which we're going to define which we're going to denote by a set of variable Z.

So now we have this set of variables Z and the question is, when can X influence

Y given evidence about Z? So the first two cases are fairly

straightforward, having evidence about Z that's not related that's not X or Y

doesn't change the ability of a variable to influence one, but to which is

directly connected. So here also,

if X is directly connected to Y in either the causal or the evidential direction,

if you tell me something about one of them, it can change my beliefs about the

other. Now, let's look at these four cases that

are that are the cases that are the most interesting ones.

That is when can X influence Y via intervening node W?

'Kay? And now there's really two cases,

either W is in my evidence set Z, oops, sorry, either it's in my evidence set Z

or it's not. So let's start with a case where W is not

in my evidence case in the evidence set Z.

Well, on this case, I didn't get to observe the W, so I'm asking whether X

can influence Y via W, and there's really no difference between

this case and the previous one. That is, for example, difficulty can

still influence letter via grade if the grade is not observed.

So here, here, and here we have exactly the same

behavior as before. That is, the intermediate variable

through which the influence flowed was not observed, and therefore,

there's no reason Y Y observing X can change things.

Before we go down to the final case, let's contrast this for these three

cases, with the case where W is observed, W is evidence.

So now, lets consider for example this tray over here where difficulty

influences the letter via grade. So this is not an edge in the Bayesian

network, this is just demonstrating the flow of

influence on [INAUDIBLE] double line. So now, the question is, we know that,

that not, that observing difficulty can change my distribution of the value of

the letter, but what if I tell you the grade?

That is, I know the student got an A in the class.

Now I'm telling you that the class is really hard,

does that change the probability distribution of the letter?

No, because we already know that the student

got an A, the letter only depends on the grade. And

so in this case, influence can't flow through grade if grade is observed.

So in this case, we have this situation what about the evidential case where

we've already talked about the fact, that evidential, that, that probabilistic

influence is symmetrical? So if difficulty can't influence letter

where grade is observed, letter can't influence difficulty when grade is

observed. And so once again, we have an no

influence in here. Finally well, not finally, but the third

case is the one where we have a common cause that has two effects.

So in this case for, for example, the SAT changing my beliefs in grade via

intelligence. And again, we know we've already examples

in fact, that the SAT can change the probability distribution in grade. But if

I tell you that the student is intelligent, then, if, then it doesn't,

there's no way for the SAT to change my probability distribution in grade.

Now, I'm giving you this as sort of a high level intuitive argument,

but it's it's possible, and we'll actually go through an argument

to demonstrate that this is really what's going on here and that these

probabilistic influences are lack there really do hold in, in a graph such as

this. Okay,

so let's talk about the last and, and most interesting case, which is the case

where we have the structure. So, this is this case over here, X, and

so for example, difficultly, can difficulty influence intelligence via

grade? And if grade is observed, this is exactly

the case that we've seen before, this is the case of intercausal reasoning

or that, that we demonstrated earlier. So in this case, if W is in Z,

actually, we're in the case where influence can flow,

so this case is working in exactly the opposite to the previous three cases.