 Hello everyone, this is Alice Gao. In this video I'm going to discuss the first example on constructing a correct Bayesian network. This is the example in lecture 11 on slide 46. In this example we're given a Bayesian network between B and A and W. It's a causal chain going from B to A to W, and then we're asked to construct another correct Bayesian network while adding the variables into the network in reverse order. So the correct answer here is that the network is still a causal chain, but let's look at how I derived this answer. Based on the variable ordering, we are going to add the nodes to the network one by one, and for each node we are going to decide on the parents parent node of that node that we're adding. Step number one, add the first node to the network. There is nothing in the network yet, so there's not much we need to do here. Step number two, we need to add a new node A to the network. Now because W is already in the network, now we need to ask ourselves a question. What are the parent nodes of A? Notice here that there could be no parent, that's possible, or there could be any number of parents depending on how many existing nodes we have in the network already. So recall the rule for choosing the parent nodes. We need to choose a set of parents from the existing nodes, and we need to make sure the set is as small as possible so that this set can separate the current node from the rest of the existing nodes in the network. So given the set of parents, the node we're adding should be independent from the other nodes in the network. So we have two possibilities here. If A is already independent from W, given nothing, basically given no parents, that means we can directly add A to the network with no parents. This is one possibility. If A is already independent from W, we don't need any parents who make A independent from the existing nodes. But on the other hand, if A is not independent from W, then we need at least W to be A's parent. Given W, A is independent from everything else, but there's no other nodes in the network anyway. These are the only two possibilities given that we only have one node in the network so far before we add A. Now which one is the correct one? Let's go back to the original network and look at it. In the original network, we have a directed edge between A and W, which means if A changes, then our belief about W would change. So these two nodes are definitely not independent. So the first option is incorrect. We need to choose W to be the parent of A. So far so good. Let's look at step number three. Now we want to add B to the network, and we already have two nodes in the network. One is W, one is A. So given that there are two nodes in the network already, we have a few more possibilities. So here are the possibilities. Let's think about these in turn. We want to minimize the number of parents, the size of the parent set. So let's start by thinking about the smallest size we can have, which is zero. If we choose the parent set to be empty set, that means that we need to verify is B independent from both W and A. If B is independent from both W and A, then we can just add B to the network, make no connections. So B has no parent and then we're done. So is B independent from W and A? If we look at the original network, of course not. B and A are directly connected. So B is not independent from A. B is also connected to W indirectly through A. So B is not independent from A. B is not independent from W. This option does not work. So having a parent set of size zero does not work. Let's now think about having a parent set of size one. So given that we have two nodes, we have two possibilities. One is we can choose A to be the only parent of B, or we can choose W to be the only parent of B. Okay, let's look at these possibilities. First of all, can we choose A to be the only parent of B? Well, here we have to verify, given A is B independent from W. If our answer is yes, then yes, we can choose A to be the only parent of B. And this would be the minimum set and we're done. So looking at the original network, the answer is yes, right? Given our original network, we already know that knowing the value of A, this chain is cut off. So B and W are independent. So given A, B is indeed independent from W, which means this is the correct one, right? You can also verify that the other one is not correct. Because so the other one requires us to verify given W is B independent of A. Well, if we know the value of W, there's still a connection between B and A in the original network, right? So B and A are still going to influence each other. This is how I got to the correct answer that I showed you, where we still get a chain going from W to A to B. That's it for this video. I will see you in the next video. Bye for now.