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. Well, we have one tricky thing left, which is what happens if W is not in Z? Now, the naive conclusion might be to say, well, you know, if the W is not observed, then it's exactly the same as before and influence can't flow so I am tempted to put an X right over here. Except that this is not quite right, because, what happen if I don't observe grade, but I actually observe letter? If 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 too, activates the V-structure, that is, it gives me in, evidence that needs to be explained, I can explain it via difficulty or via intelligence. And so and so at that point it establishes the connection, the, 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, so that there, so this X if W and all of its descendants are not observed, or conversely, that this influence can slow either, if W or, oops, [SOUND] or one of its descendants is observed. So this tells us a taxonomy of when 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 more general flow of influence. So for example, when can influence flow from S through I through G 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 is 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 through Y, but you kind of fall down the river when you hit the grade and you can't climb back up because there's nothing stopping you. And so, that too doesn't allow influence the 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 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 that's kind of an analogy. Okay, so how do we turn this into a formal definition? We have that the trail X1 up to XN and, and there should have been a K here as well, is active given Z, if, now 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 its descendants is observed. So this is activate V-structures. Now, all the other valves have to be open, so no other Xis that are not in V-structures, so Xi not in [INAUDIBLE] not in V-structures, I mean not at the sort of the next is, the bottom of the V in the V-structure. No other Xi can be observed. And that is the definition of an active trail and basically what we have is that influence can flow in the network through active trails.