WEBVTT 00:30.000 --> 00:37.360 Hey, good morning, good afternoon, and good evening everyone again at the next session of AI42 00:38.120 --> 00:43.460 And before we go into details, let me welcome my partner, Håkon. Hi, Håkon 00:43.940 --> 00:51.660 Hello, Eve. How are you? I'm good. And hi, Heine, our beautiful presenter. Hi, how are you, Heine 00:52.260 --> 00:56.880 I'm doing very well, thanks. It's going well. Awesome. That's good to hear 00:56.880 --> 01:04.480 and today we are going to show you again something awesome from the mathematics field 01:04.480 --> 01:11.640 which is going to be probability and this is another one of my favorite parts of the 01:11.640 --> 01:19.540 mathematics field and it's because it is like getting very very close to to ai and data science 01:19.540 --> 01:25.360 honey would you like to share some words about this exactly so we are going to be talking about 01:25.360 --> 01:32.360 probabilities today. And as you kind of stated, it is kind of taking us towards machine learning and being able to 01:32.500 --> 01:40.360 predict something. So it's kind of that bridge from just having data and then going towards making really kind of new 01:40.360 --> 01:45.280 information out of it. So it's going to be very exciting today 01:45.280 --> 01:48.820 I'm excited for this session. And I can't believe it's already the last one 01:50.140 --> 01:54.460 Yes, and this is somehow it's sad. But in another way, it is 01:54.460 --> 01:58.780 really cool that we have learned a lot already again. So that's exactly nice 01:58.780 --> 02:04.140 And next it's the step of really starting to implement it and use everything that has been 02:04.140 --> 02:12.860 learned. Yeah, that's very true. But before we get there, let's go, let us give you a quick intro to AI42 02:24.460 --> 02:35.580 Hi again. Hello. So let's say a few words about AI42. Would you all can start it please 02:36.500 --> 02:43.180 Yes. So the idea here with why we started AI42 is we saw that we couldn't really find one good 02:43.180 --> 02:48.620 resource where they could cover everything from starting out with the basics over to actually do 02:48.620 --> 02:54.060 some some real implementation with machine learning so then we start me and eve started 02:54.060 --> 03:00.460 up this ai422 initiative so what we do here is we will be streaming uh two times a month on 03:00.460 --> 03:06.460 wednesdays and then we will invite recognized speakers and experts that will teach you about 03:06.460 --> 03:13.180 different topics so we started started out with mathematics and probability theory and then we 03:13.180 --> 03:17.520 we will move more into different languages like R, R and Python 03:18.100 --> 03:23.960 and SQL, for example. And then we go more into actual machine learning and data science 03:24.480 --> 03:31.100 And in addition to this more theoretical part, we also will have some practical workshop sessions 03:32.960 --> 03:40.160 Yes. And we would like to do this by inviting a lot of experts 03:40.240 --> 03:45.780 basically, in the different fields so they can also give you a lot of knowledge 03:45.780 --> 03:53.320 and you can extend your network as well with the help of these people in the field of data science and AI 03:55.560 --> 04:05.940 Yes. And we are also visible here on social media, so you can follow us on Instagram or Twitter or Facebook 04:06.440 --> 04:10.660 And if you scan our QR code, you can get to both our YouTube page 04:10.660 --> 04:17.220 where you can find all our previous sessions. And we will announce our next sessions on our meetup page 04:17.220 --> 04:27.020 where you can also see on this page here. Yes. And we want to say a big, big, big thank you for our sponsors 04:27.020 --> 04:34.120 which are Microsoft and Myles. We also want to say thank you for all the contributors 04:34.120 --> 04:42.860 who helped us making this happen. and you can also learn some more if you join us now 04:42.860 --> 04:50.360 and follow us on Twitter, Facebook and Instagram. So I'm very happy that you are guys here 04:50.620 --> 04:56.580 How about getting started with some probability then? Yes, let's go back to Heineer 05:06.080 --> 05:16.920 Hi, Heini. Welcome back. Hello. So this session is going to be about probabilities 05:17.700 --> 05:22.180 And would you like to give us a quick intro about what are we going to cover today 05:23.240 --> 05:30.900 So really today's goal is to understand probabilities and give some idea of the different ways to calculate those 05:30.900 --> 05:35.520 and then at the very end I'm going to give a little 05:35.520 --> 05:39.280 kind of an overview of how all these different things we've been covering 05:39.280 --> 05:45.320 fit into doing certain things in machine learning That sounds really interesting 05:45.320 --> 05:51.620 Okay, do you have any other Yeah, go ahead, sorry I just wanted to say to the audience 05:51.620 --> 05:55.640 here that before we start, if you have any questions, just feel free to post them 05:55.640 --> 06:00.560 in the chat and then we will have a Q&A after the session is finished 06:00.560 --> 06:10.060 yes sounds good so honey are you ready to take it away yes i am very ready and let's go 06:10.060 --> 06:29.620 good evening everyone and welcome also on my behalf to the our fifth 06:29.620 --> 06:36.020 mathematics session which will be around probabilities and before we get too deep 06:36.020 --> 06:41.760 into probabilities, I just want to kind of set the stage again and kind of give you an overview of 06:41.760 --> 06:48.180 what we have been covering before and how we're now ending up into the area of probabilities 06:49.300 --> 06:56.420 So really on our previous sessions, we've been looking at this problem of us having some kind 06:56.420 --> 07:03.040 of data set. And of course, in the beginning, we just want to find some kind of information about 07:03.040 --> 07:08.140 the data that we have. Last time, we looked very extensively into statistics and found all those 07:08.140 --> 07:15.400 different methods associated to understanding our data better in terms of looking at statistics 07:16.460 --> 07:21.960 But then on the previous sessions, we've also been asking these questions that how can we 07:21.960 --> 07:38.924 find this line for our data set that best describes it And then after that what comes in is how likely is some kind of event to occur And that is really what takes us into probabilities And so what we been going through in the past sessions 07:38.924 --> 07:44.824 is we, in the first one, we looked at basic algebra, including looking at things like 07:45.544 --> 07:53.424 data sets, equations, and solving equations and such, really setting the base for all of the other 07:53.424 --> 08:00.304 math to follow. Then in the second one we were looking into calculus and calculating 08:00.304 --> 08:06.744 things as derivatives and integrals. And then in our third session, which was around linear 08:06.744 --> 08:15.604 algebra and might have been the most heavy of the sessions so far and kind of most extensive 08:15.604 --> 08:21.924 in terms of how it's used in machine learning as well. And then last week, no, two weeks 08:21.924 --> 08:31.524 ago we went into statistics and looking at those central tendencies and distribution and so forth 08:32.084 --> 08:35.844 And now in this fifth session we will be looking into probabilities 08:37.764 --> 08:42.004 So if you have not been following along to the other sessions, I'm Heine Ilmarinen 08:42.564 --> 08:51.044 I live in Finland and I work as a DevOps console at Polar Squad. I do work with Azure currently and 08:51.044 --> 08:56.704 And I also do training around Azure since I am a Microsoft Certified Trainer 08:56.704 --> 09:04.704 And my favorite topic at work is definitely architecture and figuring out architectural problems 09:04.704 --> 09:08.724 But I used to study math and actually started to become a math teacher 09:08.724 --> 09:13.764 So it's really nice that I get to combine kind of those my past skills and current skills 09:13.764 --> 09:18.984 and get to run these sessions for you around how to use mathematics for machine learning 09:18.984 --> 09:27.304 and data science and everything in that area. And as you might see in the following slides 09:27.304 --> 09:35.124 I am a bit of a dolder and like to draw pictures of everything pretty much. So we will be looking 09:35.124 --> 09:42.024 today into probabilities. We will start first. Well, what we're looking at with probabilities 09:42.024 --> 09:46.824 is really understanding the likelihood of things. We're looking at how likely is something 09:46.824 --> 09:53.484 to occur. And really, if you think about it, that is already kind of talking about things regarding 09:53.484 --> 09:59.204 being able to predict something, being able to determine whether something is likely to happen 09:59.204 --> 10:05.544 or not. And to get to that point, we're first going to start looking at basic probabilities 10:06.624 --> 10:13.404 Then we will look at, well, how do probabilities work in real life? And then that will take us 10:13.404 --> 10:19.804 into the law of large numbers. Then we will look at conditional probability 10:19.804 --> 10:27.004 and how can we determine independent versus dependent events. And then as the last point 10:27.004 --> 10:32.024 we will be looking at the Bayes' theorem. And then at the very end, as I mentioned 10:32.024 --> 10:35.944 I will be kind of tying in everything that we've been going through 10:35.944 --> 10:42.264 with the different sessions so far and the learnings that we can take from here 10:42.264 --> 10:49.384 for our machine learning career. So let us start with basic probabilities 10:49.384 --> 10:56.484 So when we're talking about probabilities, we are really looking at how likely is an event to occur 10:56.484 --> 11:01.484 And so how we would calculate that is by looking at first 11:02.624 --> 11:09.984 the number of ways that event can occur. And then we want to divide that by the total number 11:09.984 --> 11:18.304 of options, of outcomes that we have. And I think one of the most common examples 11:18.304 --> 11:24.224 used in probabilities is throwing a dice, but it's used a lot because it is so efficient 11:24.224 --> 11:31.564 to really demonstrate this point to us. So I'm not gonna lose something that works 11:31.564 --> 11:36.724 So we're just gonna take a quick example here using the role of a dice 11:36.724 --> 11:47.484 So if we think that we're rolling a dice and we want to calculate the probability of rolling a 1 11:49.824 --> 11:58.764 what we need to think, what is the ways in which we can roll that number 1? Well, we can roll it 11:58.764 --> 12:07.204 only by one way, because we only have one one in the dice. So the number of ways it can happen is 12:07.204 --> 12:13.224 one. And then the total number of outcomes in a dice is, of course, six, because there are six 12:13.224 --> 12:20.584 different numbers on that dice. So the probability is one out of six. And if we want to turn that 12:20.584 --> 12:29.464 into a percentage, we would just do that division 1 divided by 6 and multiply by 100. So we would 12:29.464 --> 12:40.904 get it to be about 16.7%. And in these simple situations, it is really easy to calculate these 12:40.904 --> 12:47.144 probabilities. But of course, our situations might be way more complex, there might be 12:47.144 --> 12:54.104 multiple events happening and things affecting one portion or the other. So we will be taking 12:54.104 --> 13:00.424 steps into those more complex situations. But this is our starting point. Just simple event 13:01.464 --> 13:06.504 nothing else affecting it, and we are just able to directly calculate the outcome 13:08.344 --> 13:12.504 But if we think about this event of rolling a dice in real life 13:12.504 --> 13:20.824 we now know that the theoretical probability that we just calculated, we just got the result 13:20.964 --> 13:27.584 we know that rolling one, the probability is this one divided by six and about 16.7% 13:27.584 --> 13:35.124 But if you think about real life situations, there is something called experimental probability 13:35.644 --> 13:40.584 Because in real life, if we decide to test this out, test this probability out 13:40.584 --> 13:48.344 we might not get the same results. So let's say that we decide that we're going to roll the dice 13:48.344 --> 13:53.904 two times. And then from those two rolls, we're going to determine what is the probability of 13:53.904 --> 14:01.644 rolling one. And if you've played any kind of game of dice, you might sometime have gotten that you've 14:01.644 --> 14:08.004 rolled the dice twice, and both times you have gotten one. So we roll it once, we get a one 14:08.004 --> 14:16.244 and we roll it twice, we get another one. And so if we have a set where we roll it 14:16.244 --> 14:26.624 two times, and both of those times we get ones. In this case, when we're calculating the probability 14:26.624 --> 14:37.104 we take the number of events where we got one, and then we divide it by the number of total events 14:37.104 --> 14:47.104 So down here it's number of total events. And at the top, it is the number of events 14:47.104 --> 15:07.609 where we got the desired outcome. So this is number of times we get one But of course on the other hand we could also in two rolls we could get no ones as well 15:07.609 --> 15:14.469 So, but does this mean that when we do this experiment, we roll the dice twice 15:14.469 --> 15:21.029 Does that mean our probability is now 100%? Does it work that way 15:21.029 --> 15:34.109 Well, not quite. So what we are to pull away from this scenario is that we have a difference of the experimental probability, and then we have a theoretical probability 15:35.429 --> 15:43.709 And what the relationship between these two is, that is what is called a law of large numbers 15:43.709 --> 16:00.029 And what this means is that it means that the more we repeat an experiment, the closer we should get to that kind of calculated results, the theoretical probability 16:00.029 --> 16:07.429 So in our example, we just rolled the dice twice. But if we keep rolling that, it is quite likely 16:07.429 --> 16:15.389 that maybe the next ones will not be dices with one. And we roll it again, and maybe we do get 16:15.389 --> 16:21.509 another one. But we roll it again, and we roll it again. And at this point, we already get to a 16:21.509 --> 16:29.029 situation where we have our probability, our experimental probability is one third, which is 16:29.029 --> 16:38.309 already much closer to our theoretical probability than was our first two tosses. So this law of large 16:38.309 --> 16:45.349 numbers really states that the more we repeat an experiment, the closer we will get to that 16:45.349 --> 16:53.269 theoretical probability. So if we were to repeat this experiment, for example, thousands of times 16:53.269 --> 17:03.469 that probability of getting a one would fall very close to that probability that we got by 17:03.469 --> 17:11.109 calculating, which was the one-sixth. It would be out of the thousand rolls of dice, about one-sixth 17:11.109 --> 17:19.669 would be ones. So really the probabilities very much tie together with an actual data set that 17:19.669 --> 17:25.969 we might have. So we might gather some data, and from our data, we might be able to calculate some 17:25.969 --> 17:33.749 probabilities. And depending on how large our data set is, that probability will be more or less 17:33.749 --> 17:40.569 accurate. And the less we have data, the less accurate that probability can be. And whereas 17:40.569 --> 17:49.209 the more we have data, the more accurate that data will be as well. And if we look at this situation 17:49.209 --> 17:54.609 that we have been looking at here. This case where we just roll some dice 17:55.529 --> 18:02.629 This particular case has been a situation where we have very discrete values 18:03.349 --> 18:14.269 So that roll of dice cannot get any value whatsoever. So it has some discreteness to what it can be showing 18:14.269 --> 18:25.429 So, sorry, I'm trying to change the slide here. There we go. So really, if we think about how then 18:25.429 --> 18:32.429 those rolls of dices will distribute, if we repeat it many times, it will create this discrete 18:32.429 --> 18:40.149 distribution where these different numbers on the dice will get a different repetition of values 18:40.149 --> 18:50.309 And those will be our probability distribution for that measurement that we're doing. For example 18:51.029 --> 18:57.469 this could describe our distribution when we're rolling a dice, except I have five bars here 18:57.569 --> 19:01.929 so we should actually have six to really be representing a roll of the dice 19:03.089 --> 19:08.189 So this is one kind of measurement where we might be looking at the probabilities 19:08.189 --> 19:16.549 But the other type that we might have is this, which is a continuous distribution 19:17.349 --> 19:26.849 So where the values that we're measuring can get any value between certain numbers, maybe some minimum and maximum numbers 19:27.249 --> 19:34.229 and then each of those values will have a different amount of repetitions again 19:34.229 --> 19:40.769 So we again have the number of repetitions and the values, and this is the continuous distribution 19:42.069 --> 19:48.209 And you might remember, we actually took a look at this in our previous session as we were looking 19:48.209 --> 19:54.109 at statistics, because we can, of course, look at the distribution of the values from the perspective 19:54.109 --> 20:00.889 of this is the data that we have gathered, or we can look at it from the perspective of what is 20:00.889 --> 20:07.909 the probability of each of those values that we might get. And of course, as we looked at how to 20:07.909 --> 20:14.049 calculate the probability, we look at the number of events per that value that we're looking at 20:14.289 --> 20:21.409 divided by the total number of events. So it will of course be related to the number of events we 20:21.409 --> 20:29.449 have for each specific measurement. So that is why just the general distribution and the 20:29.449 --> 20:35.689 probability distribution will look very alike. But as when we're talking about probability 20:35.689 --> 20:42.649 distributions, we're most often, instead of showing the number of repetitions, we instead 20:43.369 --> 20:53.449 here on this bar here, we change this to the probability of a specific value 20:53.449 --> 21:09.449 So the scale at which we represent those values will change. So we've been looking at how we can calculate the probability for one event 21:09.449 --> 21:15.449 But sometimes there is the case that we might have multiple events happening 21:15.449 --> 21:21.449 And in that case, we will need something called a conditional probability 21:21.449 --> 21:33.449 But first to understand what conditional probability is about, we need to look at whether these two events that we're looking at 21:33.449 --> 21:40.449 So we start with two events, and we need to see whether these two events are independent 21:40.449 --> 21:59.689 or whether they are dependent. So those are our two options. And what this means is that if we look at independent events 22:00.169 --> 22:07.029 then it means that if we have event A, then it does not affect the event B in any way 22:07.989 --> 22:15.909 So if we say that event A is, for example, one roll of a dice 22:18.709 --> 22:24.789 and then the event B is the second roll of a dice 22:26.869 --> 22:33.034 is the first roll going to affect what number we gonna get out of the second roll 22:35.094 --> 22:40.394 No, it's not. It's gonna, we're still gonna have the same probability 22:40.394 --> 22:49.694 So if we're now going to calculate the probability of rolling a one and a one in our two roll situation here 22:49.694 --> 22:59.054 roll situation here. So to calculate that, we need to take the probability of A 23:01.054 --> 23:09.374 and then we need to multiply that by the probability of B. So we need the probability of what is 23:11.214 --> 23:16.734 the probability of getting one in the first roll of dice, and we need the probability of B 23:16.734 --> 23:22.894 of what is the probability to get one out of that second roll of the dice. And so this means 23:22.894 --> 23:29.054 we have one sixth since we calculated that before, because we have one option out of the six to get 23:29.054 --> 23:37.374 one. And then for the second roll, the probability is exactly the same. And we just multiply those 23:37.374 --> 23:47.614 together. And so in this case, we get 1 out of 36. And again, you could write this as a percentage 23:47.614 --> 23:56.734 if you would like to. We're just gonna leave it at that for now. So why do we do the multiplication 23:58.494 --> 24:11.774 Well, because if we think about it, if you think about these two rolls of the dice, the combination of those two rolls 24:12.254 --> 24:19.774 there's no longer six options, but there is six times six options total. Because if you get a one 24:19.774 --> 24:26.174 on the first roll, you could get a one, two, three, four, five, and a six on the second roll 24:26.174 --> 24:31.694 If you get a two, you can again get one, two, three, four, five, six on the second roll. So you 24:31.694 --> 24:37.854 need to kind of multiply each of those options on the first roll with the options on the second roll 24:37.854 --> 24:43.774 And that is why we use multiplication in these scenarios. So whenever you're thinking about 24:43.774 --> 24:51.534 what is the probability of the first one and the second one, then you use multiplication there 24:51.534 --> 25:02.374 And sometimes, in a way, this can feel a little, I now have the word only in Finnish in my head 25:03.274 --> 25:14.014 counterintuitive, that's the word, to do the multiplication. But here it shows that rolling one, number one, in our roll of the dice 25:14.494 --> 25:31.054 is more probable than rolling it twice in a row. And if you think about it in that way, that makes complete sense. So multiplication when you're looking at two events. But what if we have two dependent events 25:31.054 --> 25:43.674 In that case, we can still look at this situation of having those two events and multiplying them together 25:44.014 --> 25:53.034 But in that case, we need to consider also taking into account the other scenario 25:53.034 --> 26:03.034 So, for example, if we have an event that is, it rains 26:07.034 --> 26:21.514 And we have another event that says, it is sunny. But then we also know that if it's sunny, it is a bit less likely to rain 26:22.154 --> 26:29.034 But we still do want to calculate the probability of A and B, that it is both raining and it is 26:29.034 --> 26:36.954 sunny. So to do that we cannot just take the probability of it rains and it is sunny, but we 26:36.954 --> 26:45.834 have to take the probability of it rains. And then we need to take the probability of it 26:45.834 --> 26:57.434 it is sunny when we already know that it's raining. So this second part here is telling us 26:58.714 --> 27:03.994 about, well, if A has already happened, then what is the probability of B 27:05.834 --> 27:13.994 So for example, in this scenario, we could have a probability for it rains, because right now we 27:13.994 --> 27:23.434 don't have any numbers here yet. But let's say that we have a probability of a training being 0.2 27:25.674 --> 27:37.834 And we have a probability of it being sunny, let's say 0.4. So here we see that we want our 0.2 here 27:37.834 --> 27:47.614 but actually we don't know what this probability here is. What is the probability of it being sunny 27:47.614 --> 27:57.554 if we know that it rains? So in that case, we need to have possibly some more information 27:57.554 --> 28:07.594 and maybe in that this case it has been given to us what is the probability of it then raining 28:07.594 --> 28:16.234 if it is no it being sunny when it rains so we're in this case we're gonna assume that we have been 28:16.234 --> 28:21.574 given that and let's just say that is 0.2 and we're gonna return to this in just a little bit so 28:21.574 --> 28:28.214 So we're going to go over this whole process from a different angle to make it, let it make sense 28:28.654 --> 28:35.714 So pretty much we, in this case, we were given this probability of it is sunny when we already 28:35.714 --> 28:45.694 know it's raining. We know that it is 0.2. All right. So what tells us whether those two events 28:45.694 --> 28:53.874 is R independent or dependent. Well, if they're independent, we can just take the two numbers 28:53.874 --> 29:02.954 and multiply them together directly. If they are dependent, then we need to have 29:02.954 --> 29:11.254 the probability of the second event happening when already A has happened. So in this case 29:11.254 --> 29:17.434 we have the probability of it rains times then the probability of, we know it's already raining 29:17.634 --> 29:24.234 so now the probability of it being sunny is 0.2 as well. And so we can then take those two numbers 29:24.234 --> 29:34.554 together and then calculate those and get the probability for that. So 0.2 times 0.2 means 29:34.554 --> 29:45.074 0.04 in this case. All right. So that is the first thing that we need to know here 29:47.474 --> 29:54.034 But from this calculation of, oh, we actually need to know what is the probability 29:54.034 --> 29:59.974 when one of these events has already happened. We then get to this conditional probability 29:59.974 --> 30:09.420 So we're able to take this equation that we ended up here with. So we're going to be able to take 30:09.420 --> 30:24.760 probability A and B equals probability A times probability B when A has happened. So this is 30:24.760 --> 30:49.560 really saying probability of B when A has happened. So this is where we started from 30:49.560 --> 30:58.780 So what if we don't have that information? What if we don't know what the probability of B is when A has already happened 30:59.540 --> 31:08.460 Well, then in that case, we can use the information of maybe we do know what is the probability of A and B happening together 31:08.460 --> 31:20.280 So then what we can do with this, if you remember from our basic algebra to manipulate an equation 31:20.280 --> 31:27.960 in this case we can do a division here and divide by Pa here. And that way we're able to get the 31:27.960 --> 31:48.520 result of PBA equals P and B divided by P . So that's what we get from there. So in this case 31:48.520 --> 31:54.860 if we have the probability of A and B together and we have the probability of A, then we can solve 31:54.860 --> 32:01.320 for what is the probability of B when A has already happened. And this is called conditional 32:01.320 --> 32:07.740 probability. So sometimes we, for example, might have some information that, well, we know that 32:07.740 --> 32:18.639 for somebody to, well, let's go back to our raining example. We might know the probability of 32:18.639 --> 32:25.800 it raining and being sunny at the same time, and we might know the probability of it just being 32:25.800 --> 32:33.280 sunny. And then by using those, we're able to get the probability of it raining when it is already 32:33.280 --> 32:41.340 sunny. So that's how it works together. So a lot of times we're, rather than giving the probability 32:41.340 --> 32:48.139 of when something has already happened, we most often have the probability of two 32:48.139 --> 32:54.760 happening together. So that's how we're then able to use this conditional probability to our 32:54.760 --> 33:01.160 advantage. And if we really think about that, this, most of the time, we don't have just one event 33:01.160 --> 33:06.840 we're looking at. So that is why it's quite important to be able to use this conditional 33:06.840 --> 33:16.160 probability to our advantage and use it when we might need it. Then, as kind of where we can get 33:16.160 --> 33:24.080 from here. If we look at this calculation, another way that we could have presented this 33:24.960 --> 33:35.840 same calculation here is that we would have taken P, A and B, but we could have looked at it from 33:35.840 --> 33:43.360 the other direction and taking the probability of B times the probability of P when A has already 33:43.360 --> 33:53.520 happened. And so if we take these two information that we have gotten from here, we have both this 33:53.520 --> 34:02.240 one at the top and this one at the bottom. We notice that they equal the same thing. So we're 34:02.240 --> 34:20.720 able to take both of these with just writing these two out times too many things to write 34:20.720 --> 34:30.239 here. And from here, we get to something called a Bayes' theorem. And what the Bayes' theorem 34:30.239 --> 34:35.500 is stating that when we have some kind of information about our probabilities 34:35.500 --> 34:43.860 then we can find out one other aspect based on the information that we do actually already have 34:43.860 --> 34:48.260 So based on the information we have, we're able to solve other probabilities 34:48.260 --> 34:53.360 that we might be missing. So in that this case, we're going to take this 34:53.360 --> 35:02.279 and for example, we're going to just solve it so that we can get PAB equals 35:03.500 --> 35:08.480 so again, we wanna divide by PB because there's a multiplication mark there 35:08.480 --> 35:19.620 And so that means that the left side stays put, and then we're just gonna divide it by this PB 35:19.620 --> 35:36.120 that we had here. And so this is the Bayes' theorem. And so what this is telling us is that if we have some of this other information, then we're able to solve for any missing parts that we might have here 35:36.120 --> 35:44.180 And then also by combining with this previous information here, for example, this one 35:44.860 --> 35:54.800 then we are able to leverage any of the information we have regarding our probabilities in a more 35:54.800 --> 36:02.460 complex scenario. And to make any of this make sense, I'm just going to pull this information 36:02.460 --> 36:11.420 and we're going to look at a practical example. So let's just write this out here one more time 36:11.420 --> 36:31.080 so that it's going to be there for us when we need it. Okay, so that's our Bayes' theorem 36:31.080 --> 36:47.960 And then, on the other hand, we have the information about how to calculate also this situation with kind of having the information about how two of these aspects fit together 36:47.960 --> 36:56.880 So one way to leverage these Bayes theorems is when we have probability trees. So we have some 36:56.880 --> 37:05.700 kind of event that has different probabilities at some level. So let's just do one kind of example 37:06.300 --> 37:16.680 So we have, for example, done an algorithm that looks at photos. And then by looking at those 37:16.680 --> 37:24.279 photos, it will, for example, determine if the patient has cancer. And we have determined that.. 37:24.880 --> 37:39.698 Oops just a second Let me get my pen here working again So based on this algorithm we have been able to find out that there is a 4 probability that yes 37:42.078 --> 37:51.718 there is cancer based on that photo and then 0.96 that no, there's actually no cancer 37:54.998 --> 38:07.578 But then on this side where we have it saying that, yes, it is cancer, some of these will be actually true positives 38:09.078 --> 38:15.358 So it will say that, yes, it is cancer, and it actually is cancer 38:17.178 --> 38:26.358 So let's say that this is 0.8. and then we have like a false positive 38:30.858 --> 38:36.298 So it is actually then when a test is done, it is not actually cancer 38:36.298 --> 38:41.858 And let's say that is 0.2. And then on the null side 38:41.858 --> 38:48.838 we're also going to have some probabilities. So we're again going to have kind of the true positive 38:48.838 --> 38:56.458 but in this case it will be for the no. So let's.. 39:02.038 --> 39:14.418 Well, true negative actually, that's why I got confused. Negative. And let's say that the probability for that is 0.05 39:14.418 --> 39:29.418 And then the false negative then that one will be 0.95 39:31.538 --> 39:38.838 So to really understand then probabilities, we can use the Bayes' theorem here 39:39.518 --> 39:47.418 So we can, for example, calculate that, yes, the patient has cancer 39:47.418 --> 39:54.018 when they have gotten the positive test based on these results that we have 39:56.558 --> 40:09.858 So we need to be calculating the true positives against then the true negatives plus the false 40:09.858 --> 40:16.498 negatives as well. So we kind of have to take different parts from this graph to calculate 40:16.498 --> 40:33.858 these all together. So we need to take the true positives and then calculate the true negatives 40:41.778 --> 40:55.638 and then the false positives. No, I mean the false negatives. But this looks kind of tricky 40:55.878 --> 41:02.398 So how are we actually going to go about it? So if we look at this graph on the right 41:03.058 --> 41:11.938 of course, the situation that we're looking for the probability for is, well, this branch here 41:16.498 --> 41:23.498 So the test has been positive and they actually do have cancer 41:23.498 --> 41:34.498 So that is the ones that we're looking at. So those are the scenarios that we know are the ones that we want to look at 41:34.498 --> 41:45.498 And so to calculate this kind of wanted scenarios would be to take 0.04 times 0.8 41:45.498 --> 41:56.238 but since we're looking at well the test is positive but the patient actually 41:56.238 --> 42:04.238 doesn't have cancer those are the all the scenarios so we need to add both these 42:04.238 --> 42:12.478 scenarios where it's a true positive so those are part of the set 42:12.478 --> 42:31.938 Oh, sorry, I completely miswrote this at the top. Excuse me. So this is actually the true positives plus false negatives 42:32.038 --> 42:41.998 because the false negatives will give a positive result, even though they do not actually have cancer 42:41.998 --> 42:55.598 So that's what we want to have down here. All the situations where there is a positive test result, but there isn't actually, there either is cancer or there isn't 42:55.598 --> 43:11.418 So to do that, we want to then take the 0.96 times 0.05. So I have to do now some cleaning up because 43:11.418 --> 43:20.198 I actually wrote some things wrong here. So this is the... Sorry about that. So to walk through this 43:20.198 --> 43:27.938 one more time to erase any confusion. This was meant to be so that there is actually a positive 43:27.938 --> 43:39.318 test result. That's what I was meaning with this. And this was to be a negative test result 43:39.778 --> 43:47.058 And again, we have a positive test result and a negative test result. So sorry about that 43:50.198 --> 43:56.658 So we have the, yes, there is actually cancer, no, there is not 43:56.658 --> 44:01.638 Plus then we have the layer of what have the test results actually been 44:01.638 --> 44:08.718 whether it has been positive or whether it has been negative. So those test results have been then used to confirm this 44:08.718 --> 44:15.698 So whenever we get a result of yes, so this is our algorithm here saying that yes, it is cancer 44:16.898 --> 44:23.198 This is what the algorithm says. But then we have some data, we get some results 44:23.198 --> 44:31.178 and we find out that the positive test results are the probability for that from that group is 0.8 44:31.178 --> 44:37.158 but there is still a possibility of getting a negative test results by 0.2 44:37.158 --> 44:41.558 And then even when our algorithm says that no, there is no cancer 44:41.558 --> 44:49.558 it's still possible to get a positive test result by 0.05, but a negative by 0.95 44:49.558 --> 45:04.396 So that is what we're calculating here. So we taking the rightmost branch there is the desired outcomes We want to know when these both fit together 45:05.116 --> 45:12.676 And then the ones that we want to compare this to is those who still got a positive result 45:12.676 --> 45:17.916 So both also from the no branch where there is a positive result as well 45:19.536 --> 45:28.516 And then when we have these numbers, The only thing that is left is just to calculate these out. So that's all we need to do in this case 45:29.796 --> 45:35.376 So for this one, we get, let's get back to our blue so we don't get too many colors 45:35.376 --> 45:59.776 we get 0.32 divided by 0.32 plus 0.048. And so based on this, we know that 45:59.776 --> 46:15.616 actually the probability that that result is correct is actually pretty high because it is 0.87 46:15.616 --> 46:26.696 here. So if we again present this as a probability, it is 87%. So that's how we can go about it. So 46:26.696 --> 46:33.336 But probabilities are used a lot in these kind of probability trees and things like that 46:33.336 --> 46:41.696 that we might be using. So for today, we are ending the near end of probability section 46:41.696 --> 46:47.196 And so really know that you should, there is more to know about probabilities 46:47.196 --> 46:51.196 We don't unfortunately have the time to go through every single detail 46:51.196 --> 47:00.956 I wanted you to get started on understanding the basic probabilities. And then also understanding these scenarios where we have two dependent events happening 47:00.956 --> 47:07.956 from each other. And that is where we get to this conditional probability, and then also to the Bayes' theorem 47:07.956 --> 47:16.456 as well. And so in this case, I am going to start wrapping up and just reminding you what we've been 47:16.456 --> 47:21.376 going through in the previous sessions and how this all fits together 47:21.376 --> 47:28.976 So because we've been going through all this math and we're not just going through it pointlessly 47:29.416 --> 47:35.496 it's not, well, it is fun in my opinion, but for you, it might be that you're going through 47:35.496 --> 47:40.176 the sessions to really take something to your machine learning and data science 47:40.176 --> 47:48.876 endeavors that you have ahead. So really what I want to come to here in the very end is to 47:48.876 --> 47:55.756 try to help you tie together all this. So as we've been going through these sessions 47:55.756 --> 47:59.456 if you remember, as I kind of stated in the beginning as well 48:00.516 --> 48:06.096 we have started here from basic algebra. We built on top of that with calculus 48:06.096 --> 48:10.756 Then we went into linear algebra. And then in these two last sessions 48:10.756 --> 48:17.416 we have been looking at statistics and probabilities. And these two all fit together 48:17.416 --> 48:22.956 with data science and machine learning. But if we look at data science and machine learning 48:22.956 --> 48:27.516 it can be kind of a vast area and it can be very difficult to know 48:27.516 --> 48:33.936 which of these to use where. And to help you out at least hopefully a little bit 48:33.936 --> 48:39.816 to get a sense of that is that a lot of times when we look at data science and machine learning 48:39.816 --> 48:46.676 or artificial intelligence, they might be described as these two overlapping spheres 48:46.676 --> 48:55.116 where actually the overlapping part is quite large, but there is more to data science than just the things within artificial intelligence 48:55.396 --> 49:06.516 and there's more to AI than is contained within data science. And most of the time, then machine learning is thought as a sub-area of artificial intelligence 49:06.836 --> 49:14.456 That is how it's mostly presented in a lot of sources. So that will just kind of help you set the context of all this 49:14.456 --> 49:22.956 So we've been going through all these areas. So really to be able to work within these two areas 49:22.956 --> 49:30.976 we need to have the basic algebra in place in the ground underneath all the other math. Without that 49:31.056 --> 49:35.976 it's going to get really tricky quite soon. So really try to get that foundation with basic 49:35.976 --> 49:45.156 algebra really high. And then since these areas overlap and a lot of these methods are used 49:45.156 --> 49:53.016 interchangeably and kind of across both areas, this is just kind of a best attempt at approximating 49:53.016 --> 50:01.456 So really statistics leans more towards the data science side and also, oops, wrong directions 50:01.456 --> 50:07.856 statistics leads towards the data science side, but also probabilities. But probabilities already 50:07.856 --> 50:15.776 starts to move towards also the AI and machine learning needs as well. And the more we go to the 50:15.776 --> 50:21.716 side of artificial intelligence, the more we're going to need calculus and linear algebra as well 50:22.916 --> 50:30.116 But if we look at even one area of a task that you might be doing within these areas, let's say 50:30.116 --> 50:37.516 that you're, for example, just starting to look into your data, you want to understand it, you try to find valuable insights from that data 50:38.076 --> 50:51.416 For that, you might, for example, use statistics and probabilities. On the other hand, if you're starting to do linear regression, for you to do that, you're going to need some linear algebra 50:51.836 --> 50:55.816 you're going to need some calculus, but you're also going to need some probabilities 50:56.776 --> 51:03.216 So in a lot of the cases when you're working with these areas, it is not just one of these areas that you need 51:03.716 --> 51:08.396 It is that you need pieces and portions out of many of these areas 51:09.816 --> 51:15.796 But the level at which you need to understand it depends, I would say, purely on you 51:16.236 --> 51:22.336 It will become easier to understand how things work if you do have that very deep knowledge 51:22.336 --> 51:29.476 but also we have a lot of tools that do the work for you but when it becomes really essential to 51:29.476 --> 51:35.416 know these things behind the scenes is when something goes wrong when we get some unexpected 51:35.416 --> 51:43.316 results then we might be white might have the need to have the ability to really look behind 51:43.316 --> 51:50.016 and see what has been happening to map out what has happened and why did we end up with the result 51:50.016 --> 51:56.976 we got. And of course, it might be that, yes, the unexpected result was correct, but we need to have 51:56.976 --> 52:03.376 some kind of way to verify and not just blindly trust, for example, the algorithms that we built 52:03.836 --> 52:10.756 but also we need to have some understanding of the mathematics that go to making that algorithm work 52:12.636 --> 52:18.636 So I really hope these sessions have given you a general understanding of these different topics 52:18.636 --> 52:37.915 There is definitely more to know and don feel discouraged if you feel like you don know everything yet That is completely reasonable There is more to know I hope you inspired to find out more But also don be afraid to get into the actual practical implementations and get 52:37.915 --> 52:43.555 started on those, because then looking at those practical implementations, you will get motivation 52:43.555 --> 52:49.095 to look deeper into the math as well, once you see how things work on the practical sense 52:49.095 --> 52:58.095 So I really hope you have enjoyed it, and I have really had a fun time going through these five different mathematics sessions with you 52:58.095 --> 53:16.855 Hi again 53:17.175 --> 53:21.155 Heine, you are doing great again. Thank you a lot. Thank you 53:21.955 --> 53:27.295 You're welcome. And yes, the Bayes theorem always gets me confused. Every single time 53:28.095 --> 53:37.075 And now that you say that we actually got some questions, whether you could try to work 53:37.075 --> 53:42.275 Yeah, well then look at that first. Okay. So could you give some example of how bias theorem 53:42.275 --> 53:50.755 is used in practice in machine learning and data science? So this is for example, used in decision trees 53:50.755 --> 53:57.735 when we're trying to find the best possible option for example. So that's one place where it's used 53:58.095 --> 54:02.755 And feel the two of you can feel free to add anything that comes to mind right on the spot 54:03.635 --> 54:07.535 I do have something in my mind. You know what I was thinking about this 54:07.835 --> 54:13.635 This confusion metrics, for example, because it's not just it's not just when it's decision making 54:13.635 --> 54:19.895 It is also really good at making evaluation of models. So, yeah, that's not true 54:21.055 --> 54:27.295 Yeah. Very good. about when we will talk about these things on later sessions 54:27.475 --> 54:31.115 and that's why I don't want to shoot out all the cool information now 54:32.375 --> 54:38.375 So, yeah, let's have just this, that it is useful in building machine learning algorithms 54:38.375 --> 54:46.435 and to evaluate these as well. So it has a lot of useful approaches that you can go with 54:46.435 --> 54:52.755 So, and how much statistics is it necessary to know on an everyday basis 54:55.275 --> 55:03.015 Hmm, very, very interesting question. Depends what you mean with everyday basis 55:04.675 --> 55:20.095 Yes, especially nowadays, right? Yeah, exactly. I would say just have at least some general understanding of calculating averages and the different ways to do that and what it actually tells you 55:20.695 --> 55:25.635 And I think it is very useful to also have some basic understanding of probabilities as well 55:26.115 --> 55:37.675 Because really with probabilities, I think the one most misunderstood aspect is that then we try to rely on the probability for one particular outcome 55:38.155 --> 55:44.335 where actually probabilities look at the mass of events and can really give an accurate answer 55:44.335 --> 55:53.295 for when you have multiple repetitions of that one measurement. I think also if you know some probability 55:53.515 --> 55:58.995 it can be easier to actually set up a problem and to realize is this problem actually solvable 55:58.995 --> 56:03.575 with the information that we have? Do we have enough information to solve the problem 56:04.335 --> 56:11.955 And also what you mentioned, Heine, that when things go right, it's okay, but then when things go wrong 56:12.435 --> 56:18.315 it's even more important to be able to dig down and see what is the reason why things didn't work out 56:19.035 --> 56:24.575 Exactly. And I think we can easily say, and I can say it for myself 56:24.575 --> 56:31.775 that it's very often that I try to make decisions based on the information we have 56:32.535 --> 56:38.955 And I'm not talking about the information I have only, but I'm talking about all the information I can gather all around and 56:38.955 --> 56:46.275 and including all these to, to make a decision, especially if it's something big, don't have to look far 56:46.415 --> 56:52.035 just look at the situation we have right now. So, yeah. So, um, and yeah 56:52.035 --> 56:55.855 there were a few questions about whether we could take a look again at this 56:55.855 --> 57:01.055 uh, this, uh, theorem. I think the most important part that was a bit crazy when 57:01.055 --> 57:13.895 when we talked about this true positive and true negative, And if the person who asked about it is still with us, please feel free to shoot more questions if you have any 57:13.995 --> 57:18.175 So let's take a look at that again. The last calculation. That's perfect 57:18.975 --> 57:30.035 Yes. Yes, that's a perfect clarification. Let me just roll on to the right slide here 57:30.035 --> 57:33.675 I think this is the easiest. If I just back up a little bit 57:34.195 --> 57:43.195 Sorry, oops, there we go. So, actually. Yeah, one more time, I'm just gonna 57:43.195 --> 57:53.195 Yeah, so we shared the screen as well, there we go. All right, so if we just start with our problem that we're trying to solve 57:53.195 --> 58:01.195 So our scenario is that we have this, let's say, an algorithm that is able to, from a picture 58:01.195 --> 58:08.315 picture see whether somebody has cancer or not. And then we want to know that whether this is 58:08.315 --> 58:15.515 actually accurate. So we have this test data set, we have all the results from this algorithm, but we 58:15.515 --> 58:21.755 also then have the actual test results from all the patients, whether they actually tested positive 58:21.755 --> 58:35.795 for cancer or not. So to do a little more cleanup here. So these negative, positive, negative 58:35.795 --> 58:42.955 positive, those are actually referencing to the test results. So let's just leave them like that 58:43.055 --> 58:48.575 We have a negative result and a positive result from both of these scenarios, whether there is 58:48.575 --> 58:55.415 cancer or whether there isn't. And so really, if we did want to name these with the true 58:55.415 --> 59:02.035 positives and the true negatives, this here on the right, when there is a positive test 59:02.035 --> 59:12.455 result plus our algorithm said that it's positive, this is the true positive here, actually 59:12.455 --> 59:16.615 So sorry, I got my terms mixed a little bit in the explanation 59:16.615 --> 59:21.735 And it is also getting mixed because when we are talking about cancer and such things 59:21.855 --> 59:27.195 that is, we are talking about like healthcare situations and other way around 59:27.555 --> 59:32.435 So it is positive and it's not that good, right? Yeah, exactly 59:32.435 --> 59:36.315 It's a good result. And that's why it is a bit confusing at this point 59:37.015 --> 59:42.635 Yeah, exactly. So the positive is in this case now referencing that to, yes, there is cancer 59:42.795 --> 59:48.115 This algorithm is working. So the true positive reference is to yes, the algorithm works 59:48.975 --> 59:53.215 So then this, for example, here would be the false positive here 59:55.795 --> 01:00:11.600 And then on this side, this would be a false positive. negative. And then this would be a true negative because actually they didn't have cancer 01:00:12.640 --> 01:00:20.160 So what we're calculating here is when we want to calculate what is the probability of somebody 01:00:20.160 --> 01:00:28.400 having cancer when their test result is positive, we need to look at the true positives 01:00:28.400 --> 01:00:38.080 of when they... So let me just do one more step here 01:00:40.480 --> 01:00:44.540 Let's take that color. So if you think about it this way 01:00:45.320 --> 01:00:49.540 this is our A and this is our B from here, A and B 01:00:50.660 --> 01:00:56.180 And then so for the true positives, we want to take the probability of 01:00:56.300 --> 01:01:08.740 yes, there is cancer. So that is this one here, that one. So this one. And then we want to take the probability of 01:01:08.740 --> 01:01:16.780 the test being positive when they have cancer and that we have here. So we get it over here 01:01:17.300 --> 01:01:33.866 And then the probability of B of there being positive then there we do need to take all the test results where we get positive So we need to find these ones Here a positive test result and here is a positive test result 01:01:33.866 --> 01:01:42.386 So for this one, we get first these two again, and then for this one on the left, we get these two 01:01:42.386 --> 01:01:48.426 So that's how it builds up. So that's how it's using the Bayes' theorem 01:01:48.426 --> 01:02:00.906 behind the scenes. We just worded it a little differently. Yeah. And I think now that you made circles and lines here and there, I really hope it made a bit more sense 01:02:01.506 --> 01:02:20.886 Yes, I hope so as well. Otherwise, I think since now it is over with the mathematics sessions, I think we are going to put together basically a collection of resources and some of the things Heine talked about and shown 01:02:20.886 --> 01:02:25.306 and that is going to be available at our webpage as well 01:02:25.306 --> 01:02:34.306 So you can find even more details, hopefully. And maybe some practical exercises as well 01:02:35.126 --> 01:02:39.646 Exactly. So you can play around with all the things we learned already 01:02:40.346 --> 01:02:47.306 And I think we answered the questions and we got a lot of nice feedbacks 01:02:47.306 --> 01:03:02.293 feedbacks and thank you a lot everyone for joining Maybe we can show the one that Jean wrote us Yes I hope I correctly said his name 01:03:03.273 --> 01:03:08.033 And yeah, so I also did have a lot of fun today 01:03:08.693 --> 01:03:14.213 And I have the feeling that I'm sort of going to miss these mathematical sessions a lot 01:03:14.293 --> 01:03:26.033 Yeah. Because it is actually a lot of fun. Exactly. and it is always nice to go back to the basics a bit and and refresh these things in our head as 01:03:26.033 --> 01:03:33.573 well so i really i really enjoyed it so how did you guys feel about it i thought it was really 01:03:33.573 --> 01:03:39.473 really interesting and as eve said you know we we will miss you because now we've had five sessions 01:03:39.473 --> 01:03:45.533 which has been really great exactly and thank you so much for having me for this part it was 01:03:45.533 --> 01:03:51.433 really fun for me to prepare for this and I'm glad I got to go back to the Bayes theorem to 01:03:51.433 --> 01:03:57.033 make some circles and make it clearer I was hurrying through it a little bit and I'm like 01:03:57.033 --> 01:04:06.173 never hurry that is never the answer in math no not in math at least exactly you put in some time 01:04:06.173 --> 01:04:14.033 and make sure that you enjoy it also like how did you say last time like having a walk walk in the 01:04:14.033 --> 01:04:18.073 In the park. A walk in the park, yeah. Right, that's exactly. Exactly 01:04:18.073 --> 01:04:30.880 Yeah so that how it feels every time we talk about mess And well Heidi thank you again for your time and for the awesome sessions and all the knowledge that you shared with us 01:04:30.880 --> 01:04:35.800 And I hope we can welcome you at other times as well at AI42 01:04:37.160 --> 01:04:46.240 And I think it is time to say goodbye to you. And let's start and move on to our next event 01:04:46.240 --> 01:04:51.440 which is going to be in two weeks on the 24th of March 01:04:52.100 --> 01:04:57.980 which is March. Sorry, I can't talk anymore. And that is going to be about Transact SQL 01:04:58.660 --> 01:05:02.660 So you can see we go a bit into the world of programming 01:05:02.660 --> 01:05:13.760 and learn a bit more about technologies as well. And I hope you will join us that time as well 01:05:13.760 --> 01:05:21.220 yep so with that said we just want to thank all the people who have tuned in and all of those of 01:05:21.220 --> 01:05:27.120 you who looks at this afterwards on our youtube channel and looking forward to seeing you next 01:05:27.120 --> 01:05:34.440 time yes and don't forget to follow us on twitter instagram or facebook or all of them 01:05:34.440 --> 01:05:39.760 yes yes thank you see you soon thank you