So we've seen reasoning patterns where
intuitively, at least that's how we argue,
probabilistic influence kind of starts in
one node and flows through the graph on
another node. And that might seem like,
you know, a bunch of hand waving, but it
turns out that this is actually exactly
what goes on in a Bayesian network. So, what
we're going to do now is 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 is no evidence going on and we're
just asking, can variable X influence Y.
And let's look at a few simple cases. So first, if
X and Y are connected, so say X
is a parent of Y, then, yep, pretty much
it's pretty clear cut that X can influence
Y. If Y is a child of X, we already talked
about evidential reasoning. We also saw in
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 beliefs in Y,
or about Y. More interesting are the
cases, where we have indirect influence,
between, X and Y, so, let's consider a
case, where there is an intervening
variable, W, and, let's think about that, can X
influence Y, via W? And, the first case of
the causal chain such as,
for example, the one going from
Difficulty, to Letter, via Grade, and,
we've already seen, that, in this case, X
can influence Y via W in the
example 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 here also,
we have probabilistic influence
[inaudible]. Okay. The third is a
structure that looks like this. So we have
a common cause W that has two effect 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.
Okay, so the last and most interesting one
is this case which is the case of two
causes that have a joint effect. This
place is also called a V structure, for
obvious reasons, because it's shaped like
a V. And [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, in this case, the one and only
exception in this particular case. So then
let's 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 [inaudible] 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
in, 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 a
V estructure because that is the one case
where we have with no influence flows as
an example that we showed before. So this
is a block in the trail. Now let's 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 as a
set of variables Z, so now we have this
set of variable Z and the question is when
can X influence Y given evidence about Z
so the first two cases are fairly straight
forward, having evidence about Z that's
not related that's not either X or Y
doesn't change the variability of a
variable to influence one which is not
directly connected, so here also. If x is
directly connected to y and either of 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,
the later of the cases that are the most
interesting ones. That is one where  X can
influence Y through an intervening node W. And
others really two cases either w is in my
evidence set z. Opps, sorry, either it's in
my evidence set z or it's not. Now, lets
start with a case where w's not in my
evidence case in my evidence set z. Well,
in this case, I didn't get to observe 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 grade is not
observed. So here, here, and here, we have
exactly the same, behavior as before. That
is, the intermediate variables through
which the influence flowed was not
observed, and therefore, there's no
reason, why, While observing x can change
things. Before we go down to the final
case, let's contrast this for these three
cases with a case where w is observed. W
is evidence. So now let's consider for
example, this tray over here where
difficulty influences the letter by a
grade. So this is not an edge in the
Bayesian network, this is just
demonstrating the flow of influence on
[inaudible] double lines. So now the
question is we know that's, that, that
observing difficulty can change my
distribution in the value of the letter.
But what if I tell you the grade, that is
I know that a student got an A in the
class. Now I'm telling you that the class
is really hard, does that change the
probability of distribution of the letter?
No because we already know the student got
an A the letter only depends on the grade
and so in this case insolence can't flow
through grade if grade is observed, so in
this case, we have, this situation. What
about the evidential case? Well, we've
already talked about the fact that
evidential, 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 no influence 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 take for example
the SAT changing my beliefs in grade via
intelligence. And again we know we've
already seen examples in fact that the SAT
can change the probability of
distributioning grade but if I tell you
that the student is intelligent then, 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, probalistic influences
[inaudible] really do hold, in, in a graph
such as this. Okay so let's talk about the
last and most interesting case which is
the case where we have a V structure so
this is this case over here X and so for
example difficulty can difficulty
insolence intelligence via grade. And if
grade is observed, this is exactly
the case that we've seen before. This is
the case of inter-causal 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. [inaudible] we
have one tricky thing left, which is, what
happens if W is not in Z? Now, the
main conclusion might be to say, well, you
know, if W's not observed, than it's
exactly the same as before. And influence
can't flow, so I'm tempted to put an X
right over here. Except, that this is not
quite right. Because what happens if I
don't observe grade, but I actually
observe. Letter. I don't observe the grade
directly but I observe something that
gives me a strong indication of what value
the grade took. In this case this tool
activates the v structure. That is it
gives me in evidence that needs to be
explained. I can explain it by difficulty
or via intelligence and so and so that
point it establishes the connection, the
correlation between them. So that
observing one does influence the other.
And so this one is not actually quite
right. What you'd actually like to say is
that if w, cause of their. So this is X
if. W and R all the way to the descendents.
Are not observed. Or conversely that this
influences control either, if W or
one of its descendants is observed. So
this tells us a taxonomy of how influence
can flow through an intervening variable
and now we can take that and we can put it
together to define an overall model, of a
more general flow of influence. So for
example, one can influence flow from S
through I. Og into D. Well? Let's look at
a couple cases. What if I is observed?
Well, if I is observed, then it blocks the
trail. And if it blocks the trail, you're,
you're, there's no more opportunity for it
to flow. So that's a no. What about if I
is not observed but nothing else is
observed? I not observed. Ignores anything
else. Well. What happens then? Well, you
can climb up [inaudible], but you kinda
fall down the river when you hit the
grade. And you can't climb back up, 'cause
there's nothing stopping you. And so that
two doesn't allow influence to flow. On
the other hand, if I is not observed. And
G is observed. Well, now you can climb up
here, continue through here. Grade is
observed so the water can kind of go back
up the hill into difficulty. So you can
think of it as a flow, sort of a flow of
water except that different nodes behave
differently in terms of the valve
structure. So here if you observe a
variable, it closes valves that go like
this. But if you have a V structure,
closing this valve actually lets the water
climb back upstream, so it's kind of an
analogy. So how do we turn this into a
formal definition? We have [inaudible]
trail, X1 of the XN. And I have this,
should have been a K here as well. Is
active given the if and we have two cases.
First every V structure needs to be
activated and the only way we can activate
a V structure is if XI or one of it's
decendants is observed. Now this is
activate. V structures. Now all the other
valves have to be open so no other X I
that are not on these structures so X I
not in. Whe-, whereby v, not in v
structures, I mean not at the, sort of,
the nexus. The bottom of the v in the v
structure. No other Xi can be observed.
And that's the definition of an active
trail. And basically what we have, is that
influence can flow in the network through
active trails.