Now, we start to considering some very famous thing, which is called Bayes' Theorem. Actually, like a lot of things we discussed now is something trivial. But it's very popular and very important philosophically. It's somehow, say something about reasoning, what kind of probabilistic reasoning is good. So that is a special name bias and statistic or bias and argument or whatever. But the formula is very simple, let's go to it. But still I want to start with a real life example, it's real. Actually, several weeks ago when I prepared the slide, I looked in my mail. And I found that I have a message from Bank of America. But strangely this message has an address in Japan. You see this is Japan. And I thought that it's probably a scam. And why, why, what are the reasons for that? And the reasons are the following. First, I know by experience that many scam messages have foreign address. So, because these people don't want to be caught, so it's convenient for them to use the address in a foreign country. So it makes the search for who rarely use the sellers more difficult, anyway it's just an observation. Another observation just in general, if you look at all messengers, this foreigner are not so common things. It's not often, it's some clear contradiction between the place when the message comes from the name of the person and the location of the person and the address. And another third observation unfortunate, that the scam messages are rather often. So people try to somehow to prevent them or to prosecute the people who send them. But still there are quite often, unfortunately. And this three reasons are just three component of Bayesian reasoning. I will tell you, try to explain what do I mean on the next slide. Yeah, so there are three components of Bayes' reasoning. And here is the formula. So it deals with two events. They are called hypothesis and evidence, but just any two events are okay, just the names of this. But of course, they should be the same probability space. So for example, we can consider the event H and E. So, I recall the definition of conditional probability. So the conditional probability of H under the condition E, probability of hypothesis, it's somehow counterintuitive. Probability of hypothesis under the condition that the event happens is by definition the reflection of this together divided by probability of the condition. And then we can use the same formula. The definition of conditional probability in other direction. We can start with H until the other conditional probability, you see this in different order. Okay, so what? We can re-write it even like this. And it's the real computation, but it has some meaning. And the meaning is, can be explained like this. So we are comparing the probability of H with condition E, or without this condition. And with condition E, somehow it multiplies the probability without the condition by some factor. And this factor shows how much E becomes more probable if we add the condition H, so the condition multiplies probability by some factor which is another ration. So, it's kind of explanation of this factor. We'll see this an example. So let's compare our theory and our practice, our example with this e-mail. So, this is the bias formula, and what is the interpretation? So H, the hypothesis, is that the message is a scam. And the observed event is that message use a foreign address. So, how probable is that the message is scam assuming that it has a foreign address? And my reasoning was that this probability is rather high because first this thing is quite high, which means that many many scam messages use foreign address. And this is pretty low that not many messages in general have foreign address. And this is not very low, nowadays, there is a lot of scam. So we have, this is rather high, this is low, and this is not very low. So, I conclude that the entire fingers high, and this is what the reasoning why I decided that this is a scam. So, I tried to write this. So just in one sentence, the reasoning is written here. So foreign address makes the scam hypothesis much more probable, much more probable compared to original probability, because it appears in scam messages much more often than in general. So this is a reading of bias formula for our special case. And let's now look on our example with disease when somehow similar reasoning doesn't work, why the formula doesn't give things. So again, we have the disease in test. The probability of disease for test positive people. And then we have this computation, so probability of test for people having this disease is 0.9. The probability of having the positive test, we compute it using the formula of total probability, is this. And here is the probability of disease. And we get, when we compute things, we get only 8%. So why don't we get such a serious big number? And indeed what can be said, this probability of test for people having this is much bigger than the probability than the general population. But this probability of the disease is small, very small. So even the product is not very large, it's only one over twelve, actually. So, here the hypothesis was so improbable by itself that even the increase in the probability because of the Bayes' Theorem, doesn't make it very probable. This is what happens in our example. Okay, Coursera wants all the time wants to have take-home messages and repeating the main learning objective. Okay, so here the learning objective is to understand the meaning of Bayes' formula and see the cases when you apply this reasoning in the real life. So this is the message, if some conditions significantly increases the probability of some events by some big factor, K or, of course it's true for any factor but it's just if, it's interesting for big factors. So if condition B increases the probability of A, then symmetrically, condition A increases the probability of B by the same factor. So this is take-home message from Mr Bayes'.