 Hello everyone, this is Alice Gao. In this video, I am going to discuss conditional independence and unconditional independence relationships in a Bayesian network. In the previous video, I talked about two ways of understanding a Bayesian network. First, a Bayesian network represents the joint probability distribution. We have enough information to recover the entire joint distribution. Second, a Bayesian network encodes a lot of unconditional and conditional independence relationships among the variables. I'll show you three examples in this video. These are all taken from the Holmes network, so each example is fairly small, only containing three random variables. Yet, I chose these examples very very carefully because they represent the most basic and fundamental relationships we can see in the Bayesian network. In the first example, let's look at the relationships between burglary, alarm, and Watson. Intuitively, their relationship is like a causal chain. If a burglary is happening, this may cause the alarm to go off, which in turn may cause Dr. Watson to call Mr. Holmes. For each example, I'm going to ask you two questions. The first one is unconditional independence, and the second one is unconditional independence. In this video, I'm only going to give you the correct answers as well as some intuitive explanations of the answers, but in a separate video, I will include more formal verification of the answers by calculating probabilities. Here's the first question for the first example. If we consider burglary and Watson in this causal chain, are burglary and Watson unconditionally independent or not? Pause the video, think about this yourself, and then keep watching. The correct answer to this question is no. Burglary and Watson are not independent. To get this answer, we need to ask the following question. If we learn the value of burglary, would this influence our belief about the value of Watson? Intuitively, the answer is yes. If the burglary is happening, if we know that the burglary is happening, then we will believe that the probability that alarm is going off is higher, it would increase, and therefore the probability that Dr. Watson is calling Mr. Holmes would increase as well. Therefore, because burglary and Watson would affect each other, they would change our belief about the other variable so they're not independent. Question number two, are burglary and Watson conditionally independent given alarm? Again, think about this yourself, and then keep watching for the answer. The correct answer to this question is yes. Once we know alarm, burglary and Watson become independent. The network we're seeing here makes an important assumption. By only having an edge from burglary to alarm and another edge from alarm to Watson, we are assuming that Watson does not directly observe burglary. If Watson observed burglary, we would have an edge from burglary directly to Watson as well, but we don't have that. So if Watson does not directly observe burglary, Watson only observes alarm, so burglary cannot directly influence Watson. Burglary could only influence Watson through alarm. Because of this, we can intuitively reason as follows. If we know whether the alarm is going off or not, so we know whether alarm is true or false, our belief about Watson is fixed given that. Because Watson only depends on alarm. So when we know alarm, it is as if we cut this chain in the middle. And once we've cut this chain in the middle, burglary and Watson cannot affect each other anymore. So they are conditionally independent if we know alarm. Here's the second example. Let's look at the relationships between alarm, Watson and given. So intuitively, in this example, alarm could cause Dr. Watson to call Mr. Holmes. Alarm could also cause Mrs. Given to call Mr. Holmes. In this example, let's look at the relationship between Watson and Given. So first question, are Watson and Given unconditionally independent from each other? Think about this yourself and then keep watching for the answer. The correct answer is no. Watson and Given are not independent in this example. Similar to before, we need to ask the following question. If we learn the value of Watson, whether Watson is calling or not, does this influence our belief about Given? Well, if Watson is calling, then it is more likely that alarm is going off, right? Because alarm and Watson, they influence each other. Now if our belief for the alarm going off increases, then it's more likely that Mrs. Given is calling as well. So from this reasoning, we can say that if we learn the value of Watson, then our belief about Given would change. So Watson and Given are not independent. Question number two, are Watson and Given conditionally independent given alarm? Take a moment, think about this, and then keep watching. The correct answer is yes. Given alarm, Watson and Given are conditionally independent of each other. Intuitively, we can think about alarm as an event we're interested in, and then we can think about Watson and Given as noisy sensors for the event. The value of each sensor depends entirely on the event and depends on nothing else. So if we think about it this way, then if we know whether the event is happening or not, then the value of each sensor is determined, is fixed, and the two sensors cannot affect each other in any way. That's why Given alarm, Watson and Given must be independent because they cannot affect each other anymore. Let's look at a third example. This one is about earthquake, burglary, and alarm, and the relationship among these three variables. For this example, the alarm can go off for two reasons. One is that an earthquake is happening, and the other reason is that a burglary is happening. In other words, earthquake and burglary are two possible causes for the alarm to go off. For the next two questions, we are going to look at a relationship between earthquake and burglary. Here's question number one. Are earthquake and burglary independent from each other? Think about this for a moment and then keep watching for the answer. The correct answer here is yes. Earthquake and burglary are independent. If we learned whether earthquake is happening or not, this does not change our belief about burglary and vice versa. You may argue that this is not the case in real life. In real life, during an earthquake, looting may be more frequent. So these two are not independent. However, let's assume that these two are independent in the home story. Let's say this is a simplifying assumption we'll make to model the home scenario. Here's the second question for the third example. Are earthquake and burglary conditionally independent given alarm? This may be the trickiest question out of all the questions in this video. So take your time. Think about this carefully and then keep watching for the answer. The correct answer here is no. If we know alarm, then earthquake and burglary are not independent. So they're not conditionally independent given alarm. Earthquake and burglary can both cause the alarm to go off. So here's the intuition for why they become dependent if we know alarm. Suppose the alarm is going off. Given this, if an earthquake is happening, then it is less likely that alarm is caused by a burglary. So it is less likely that a burglary is happening as well. On the other hand, if an earthquake is not happening and alarm is going off, then it becomes more likely that a burglary is happening, which causes alarm to go off. So earthquake and burglary are both possible causes for the alarm. If one is happening, that means the other one is less likely to be happening and vice versa. Sometimes this effect is also called the explaining away effect. That's everything for this video. Let me summarize. After watching this video, you should be able to do the following. For each of the three key patient network structure I've talked about, you should be able to identify the conditional independence and unconditional independence relationships between the variables and also discuss their intuitions. Thank you very much for watching. I will see you in the next video. Bye for now.