 Hello everyone, this is Alice Gao. In the previous video, I introduced the concept of de-separation. This is a very powerful concept. Given any two variables x and y in the Bayesian network, we can use de-separation to test whether x and y are conditionally independent given that we observe a third set of variables e. In this video, let's look at a couple of examples where we apply this concept to determine some independent or conditional independence relationships. All of the questions in this video use the same example, same Bayesian network. So we have that if we travel on the subway or if we take a trip to an exotic area, then we might catch certain diseases like malaria or flu and these diseases may cause different symptoms such as aches on our body, fever and jaundice and fever will cause us to have a high temperature. Let's look at independent relationships between some variables. So for each question, I will tell you the answer first and then I will discuss the reasoning behind the answers. Make sure you pause the video and do the question yourself before you keep watching for the discussion. Here's question 1a, we are interested in travel subway, the relationship between travel subway and high temperature and let's assume that we observe nothing. So given that we observe nothing, our travel subway and high temperature independent or unconditionally independent. Think about this yourself and then keep watching for the answer. The correct answer here is no, travel subway and high temperature are not independent. Here's how I derived the answer. So in order to figure out the relationships of the two variables, I need to first find all the path that can connect these two variables. There's only one path in our Bayesian network. So going from travel subway to flu, then to fever and then to high temperature. This is the path. So being the only path, we need to look at this path and consider every variable on that path. And for every variable, we need to decide, determine whether that variable blocks the path or not. So first of all, does the variable flu block this path? Well, the links around flu is the first structure that we looked at, right? The first rule. So we need to apply rule number one. And rule number one says that if flu is observed, then it blocks the path. If it's not observed, then it does not block the path. In this case, we'll observe nothing. So flu does not block the path. Imagine that we can go through flu. The path is connected there. It's similar for fever. Again, the structure around fever is a chain. And then according to rule number one, it does not block the path because we do not observe fever. So the entire path is connected and not blocked. Therefore, the two variables are not independent. Question number two, or question one B, consider the same two variables travel and subway, except in this case, we observe flu. So given flu are travel subway and high temperature independent or not. Think about this yourself and then keep watching for the answer. The correct answer is that these two variables are conditionally independent given flu. Again, we can get this by you applying rule one twice. So the connection between the two variables is again this chain. And now if we look at the variable flu, we observe the value of flu. So it does block this path. And we can stop right there. Because as soon as we find a variable that blocks the path, the path is severed and there are no other paths connecting these two variables. Therefore, these two variables must be independent from each other. If you want to verify for fever is the same as before because we do not observe fever. So fever does not block the path. Question one C, this question is again asking the relationship between travel subway and high temperature except that we not observe aches. I'm going to leave the reasoning as an exercise for you, but I will tell you the answer right here. The correct answer is no. Travel subway and high temperature are not conditionally independent given aches. The key here is that again we are going to apply rule one twice to figure out whether any node blocks the path between the two variables. And also it's important to realize that observing aches does not affect anything between these two variables. Let's keep going. The next question to a, we are finally looking at the different set of variables. So we're looking at the relationship between aches and high temperature. And in this question, we observe nothing. So are aches and high temperature independent from each other if we observe nothing? Think about this yourself and then keep watching for the answer. The correct answer is no. These two variables are not independent. To see why we need to look at the path between the two and this path go through flu and fever. So for each of flu and fever, we need to apply one of the three rules to decide whether it blocks the path for flu. The links around flu correspond to the second rule, rule number two. And if we apply the second rule, it says that if we observe the node, if we observe flu, then it blocks the path. But if we don't observe flu, then it does not block the path. In this case, we do not observe flu. So it does not block the path. There's a connection through the node. And for fever, it's same as before, we will apply rule number one. And because we do not observe fever, so it does not block the path either. So this path is connected the whole way from aches to high temperature, which means the two variables are not independent. Question to be, we are still considering aches and high temperature, except that now we observe something. We observe the value of flu. So given flu are aches and high temperature conditionally independent or not. Think about this yourself and then keep watching for the answer. The correct answer is yes aches and high temperature are conditionally independent given flu. Again, we need to decide whether this path is blocked by any node on the path. The path is not blocked by fever. And this is based on rule number one. But the path is blocked by flu. And this is based on rule number two, right? Rule number two says if flu is observed, then it blocks the path. Since this is the only path between the two variables and it's blocked, so the two variables are conditionally independent given flu. Okay, we have two questions left. Question 3a, are flu and exotic trip conditionally independent? I should strike the conditionally word here because we don't observe anything. Think about this yourself and then keep watching for the answer. The correct answer here is yes. The two variables flu and exotic trip are independent from each other. Let's look at why. So there is one path connecting these two variables and this path goes through malaria and fever. So for malaria, the structure around it looks like rule number one. It's a chain and we do not observe malaria, which means that malaria does not block the path. We can go through the node and it connects the two nodes around it. Now let's look at fever. The structure around fever looks like rule number three, where we have flu and malaria, they jointly cause fever. So applying rule number three, what does rule number three say? The rule says we need to look at the node and we need to look at all of its descendants. In this case, it only has one descendant, which is high temperature. So among the node itself and all of its descendants, if all of the nodes are not observed, then this blocks the path. If at least one node in this set is observed, then it does not block the path. Well, in this case, nothing is observed. Fever is not observed and high temperature is also not observed. So in fact, they block this path. So because this path is blocked and this is the only path connecting flu and exotic trip, so flu and exotic trip are independent. Here's the final question. We are considering again, flu and exotic trip and in this case, we observe high temperature. Given that our flu and exotic trip conditionally independent or not. Think about this yourself and then keep watching for the answer. The correct answer is no, flu and exotic trip are not conditionally independent given high temperature. Let's look at why. Same as the previous question, there is one path connecting flu and exotic trip like so and same as before, we need to look at whether malaria blocks it and whether fever or any of its descendants blocks it. So malaria, the structure around it, we apply rule number one again and it does not block the path. We can go through there. Now the tricky part is with respect to fever and its descendants. So again, rule number three says that if we observe none of the node fever and also none of its descendants, then it blocks the path. But if we observe at least one of the node itself or one of its descendants, then they do not block the path. They actually connect the path. In this case, we do not observe fever itself, but we do observe the only child of fever, which is high temperature. Therefore, fever does not block the path and actually connects it. So after applying rule number one and rule number three, we find that the entire path is connected, nothing blocks it. Therefore, the two variables flu and exotic trip are not conditionally independent, given high temperature. That's everything for this video. After watching this video, you should be able to do the following. Given any patient network, pick any two variables and then pick a third set of variables to be the observed variables. You should be able to apply the separation to determine whether the two variables are conditionally independent or not, given the third set of variables. Thank you very much for watching. I will see you in the next video. Bye for now.