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So we've seen reasoning patterns where
intuitively, at least that's how we argue,

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probabilistic influence kind of starts in
one node and flows through the graph on

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another node. And that might seem like,
you know, a bunch of hand waving, but it

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turns out that this is actually exactly
what goes on in a Bayesian network. So, what

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we're going to do now is we're going to make
this argument much more rigorous by trying

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to understand exactly when one variable X
can influence another variable Y, and

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we're going to start with a case where
there is no evidence going on and we're

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just asking, can variable X influence Y.
And let's look at a few simple cases. So first, if

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X and Y are connected, so say X
is a parent of Y, then, yep, pretty much

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it's pretty clear cut that X can influence
Y. If Y is a child of X, we already talked

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

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that observing X can change my probability
distribution of Y. That's what I mean by

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influence, influence means beliefs in Y,
or about Y. More interesting are the

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cases, where we have indirect influence,
between, X and Y, so, let's consider a

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case, where there is an intervening
variable, W, and, let's think about that, can X

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influence Y, via W? And, the first case of
the causal chain such as,

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for example, the one going from
Difficulty, to Letter, via Grade, and,

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

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example that we saw. This is exactly the
same idea, except that we're going

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evidential and as we'll see in general,
probabilistic influence is symmetrical.

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That is, if X can influence Y, Y can
influence X. And so, that's here also,

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we have probabilistic influence
[inaudible]. Okay. The third is a

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structure that looks like this. So we have
a common cause W that has two effect X and

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Y. And again, it seems to make sense that
if, we observe the value of the SAT, then,

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that changes my beliefs in the student's
intelligence, and subsequently, my

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probability distribution over their grade.
Okay, so the last and most interesting one

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

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place is also called a V structure, for
obvious reasons, because it's shaped like

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a V. And [inaudible], remember, I haven't
given you any information. The question

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is, if I tell you that a student took a
class, and the class is difficult, does

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that tell me anything about the student's
intelligence? And the answer is no. And so

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this, in this case, the one and only
exception in this particular case. So then

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let's define this notion of active trail
in the context of no evidence. So a trail

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in general, is a sequence of nodes that
are connected to each other by single

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edges in the graph. So X1 up to actually
we should make this XK, so as not to

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confuse with [inaudible] variables. And
the fact that these edges are undirected

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means that they can go in either
direction. So I'm not stipulating that it

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goes up or down. So basically we saw that
in, that influence can flow from one

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variable to another variable, in the
graph. And what this definition basically

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says is that the, is that the, this
influence can continue to flow. So if it

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flows from one variable to the next, to
the next, to the next, to the next, that

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still defines an active trail. The only
thing that blocks an active trail is a

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V estructure because that is the one case
where we have with no influence flows as

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an example that we showed before. So this
is a block in the trail. Now let's look

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at a more interesting case, now we have
some set of observations which we're going

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to define which we're going to denote as a
set of variables Z, so now we have this

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set of variable Z and the question is when
can X influence Y given evidence about Z

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so the first two cases are fairly straight
forward, having evidence about Z that's

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not related that's not either X or Y
doesn't change the variability of a

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variable to influence one which is not
directly connected, so here also. If x is

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directly connected to y and either of the
causal or the evidential direction if you

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tell me something about one of them it can
change my beliefs about the other. Now

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let's look at these four cases that are,
the later of the cases that are the most

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interesting ones. That is one where  X can
influence Y through an intervening node W. And

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others really two cases either w is in my
evidence set z. Opps, sorry, either it's in

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my evidence set z or it's not. Now, lets
start with a case where w's not in my

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evidence case in my evidence set z. Well,
in this case, I didn't get to observe w.

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So, I'm asking whether x can influence y
via w, and there's really no difference

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

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influence letter via grade if grade is not
observed. So here, here, and here, we have

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exactly the same, behavior as before. That
is, the intermediate variables through

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which the influence flowed was not
observed, and therefore, there's no

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reason, why, While observing x can change
things. Before we go down to the final

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case, let's contrast this for these three
cases with a case where w is observed. W

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is evidence. So now let's consider for
example, this tray over here where

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difficulty influences the letter by a
grade. So this is not an edge in the

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Bayesian network, this is just
demonstrating the flow of influence on

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[inaudible] double lines. So now the
question is we know that's, that, that

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observing difficulty can change my
distribution in the value of the letter.

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But what if I tell you the grade, that is
I know that a student got an A in the

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

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probability of distribution of the letter?
No because we already know the student got

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an A the letter only depends on the grade
and so in this case insolence can't flow

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through grade if grade is observed, so in
this case, we have, this situation. What

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about the evidential case? Well, we've
already talked about the fact that

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evidential, that probabilistic influence
is symmetrical. So if difficulty can't

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

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grade is observed. And so once again, we
have no influence here. Finally well not

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finally but the third case is the one
where we have a common cause that has two

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effects so in this case take for example
the SAT changing my beliefs in grade via

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intelligence. And again we know we've
already seen examples in fact that the SAT

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can change the probability of
distributioning grade but if I tell you

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that the student is intelligent then, then
it doesn't, there's no way for the SAT to

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change my probability distribution in
grade. Now, I'm giving you this as sort of

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a high level intuitive argument. But it's,
it's possible, and we'll actually go

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through an argument to demonstrate that,
this is really what's going on here. And

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that these, probalistic influences
[inaudible] really do hold, in, in a graph

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such as this. Okay so let's talk about the
last and most interesting case which is

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the case where we have a V structure so
this is this case over here X and so for

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example difficulty can difficulty
insolence intelligence via grade. And if

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grade is observed, this is exactly
the case that we've seen before. This is

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the case of inter-causal reasoning or,
that, that, we demonstrated earlier. So in

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this case if W is in Z, actually we're in
the case where influence can flow, so this

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case is working in exactly the opposite to
the previous three cases. [inaudible] we

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have one tricky thing left, which is, what
happens if W is not in Z? Now, the

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main conclusion might be to say, well, you
know, if W's not observed, than it's

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exactly the same as before. And influence
can't flow, so I'm tempted to put an X

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right over here. Except, that this is not
quite right. Because what happens if I

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don't observe grade, but I actually
observe. Letter. I don't observe the grade

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directly but I observe something that
gives me a strong indication of what value

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the grade took. In this case this tool
activates the v structure. That is it

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gives me in evidence that needs to be
explained. I can explain it by difficulty

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or via intelligence and so and so that
point it establishes the connection, the

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correlation between them. So that
observing one does influence the other.

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And so this one is not actually quite
right. What you'd actually like to say is

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that if w, cause of their. So this is X
if. W and R all the way to the descendents.

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Are not observed. Or conversely that this
influences control either, if W or

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one of its descendants is observed. So
this tells us a taxonomy of how influence

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can flow through an intervening variable
and now we can take that and we can put it

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together to define an overall model, of a
more general flow of influence. So for

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example, one can influence flow from S
through I. Og into D. Well? Let's look at

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a couple cases. What if I is observed?
Well, if I is observed, then it blocks the

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trail. And if it blocks the trail, you're,
you're, there's no more opportunity for it

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to flow. So that's a no. What about if I
is not observed but nothing else is

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observed? I not observed. Ignores anything
else. Well. What happens then? Well, you

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can climb up [inaudible], but you kinda
fall down the river when you hit the

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grade. And you can't climb back up, 'cause
there's nothing stopping you. And so that

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two doesn't allow influence to flow. On
the other hand, if I is not observed. And

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G is observed. Well, now you can climb up
here, continue through here. Grade is

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observed so the water can kind of go back
up the hill into difficulty. So you can

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think of it as a flow, sort of a flow of
water except that different nodes behave

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differently in terms of the valve
structure. So here if you observe a

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variable, it closes valves that go like
this. But if you have a V structure,

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closing this valve actually lets the water
climb back upstream, so it's kind of an

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analogy. So how do we turn this into a
formal definition? We have [inaudible]

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trail, X1 of the XN. And I have this,
should have been a K here as well. Is

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active given the if and we have two cases.
First every V structure needs to be

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activated and the only way we can activate
a V structure is if XI or one of it's

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decendants is observed. Now this is
activate. V structures. Now all the other

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valves have to be open so no other X I
that are not on these structures so X I

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not in. Whe-, whereby v, not in v
structures, I mean not at the, sort of,

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the nexus. The bottom of the v in the v
structure. No other Xi can be observed.

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And that's the definition of an active
trail. And basically what we have, is that

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influence can flow in the network through
active trails.