I will continue. Our ultimate goal is to understand the probabilities that arise from machine learning and deep learning. We are conducting a crash course to learn this content in a short period of time. We will focus on the aspects of probability that are necessary for machine learning, not so much on having a strong mathematical foundation, but rather on capturing the big picture. However, no matter how much we try to see the big picture, some theoretical basis is needed. So, you can think of it as proceeding from that perspective. 

Now, how is probability defined mathematically? We have objects in the real world, and we are mathematically modeling those objects. Thus, there exists something in the mathematical world. Usually, when we talk about probability, we refer to how probability works in everyday life.

Probability refers to the likelihood of an event occurring, such as the chance of rain tomorrow. There is also a corresponding mathematical theory related to this concept. We are discussing that mathematical theory now. So, what does this mathematical theory entail? There is a set known as the sample space, which contains subsets. The sizes of these subsets can be measured, for example, one subset might have a size of 0.3, another 0.2, and yet another 0.4. This is how probability is expressed by measuring the sizes of subsets. On the page, there was a sample space, and within that sample space, there was an event that gathered the subsets. Each element of the event corresponds to the events that are collected from the sample space.

It's a subset, and it's a set of events. The probability function assigns a number to each event, but it cannot be assigned arbitrarily. There are rules that must be followed for the assignment. The first rule is what we call axioms, which are essential rules that must be adhered to. Once we satisfy these three conditions, we can move on to the next discussion. 

The first condition is that the size must be between 0 and 1. The highest probability is 1. The second condition is that the total size of the set is 1. The third condition, which may seem a bit lengthy, ultimately states that there is no intersection. Therefore, the size of the union of A1 and A2 is equal to the sum of their sizes when there is no overlap. This means that when measuring size, we are considering the size of the set, and the two sets do not intersect.

Then, the size of the union seems to be equal to the sum of each individual size, right? The discussion begins when three conditions are satisfied. So here, we are using three axioms to derive the conclusion. For example, one conclusion is that the size of the empty set is 0. To prove that the size of the empty set is 0, we use the axioms. First, we consider that the union of the empty set with itself is still the empty set, right? So far, that's okay. However, the empty set has no intersection. Therefore, according to Axiom 3, this holds true here. And the fact that this equals 1 is stated in Axiom 2. So, since it was 1 according to Axiom 2, it becomes 1 here. Starting from here, expressing it in a way that corresponds 1 to 1 with the earlier ones might be a bit easier to interpret in terms of the axioms.

Anyway, there is no mathematical problem. So, since this is 1 and this is also 1, what does that mean? This must be 0. Therefore, this has been proven. Likewise, the size of the complement set must be at least 1 minus the size of set A and B, if they are included. So, well, you don't need to go too deeply into the mathematics, but when measuring the size of a set, there are three axioms: the size of a set is 0 or greater; the size of the empty set is 1; and if two sets have no intersection, then the size of the union of the entire set is equal to the sum of the sizes of each subset. The size of the union is equal to the sum of the sizes when there is no intersection. So, these things must hold true regarding probability, for example.

When we want to find a corresponding model for a natural phenomenon or something similar, we need a model that includes a sample space, a set of events, and probabilities. So, we need to prepare how to find the events, how to create events, and how to assign probabilities. Only when we have these three components can we apply theories about probabilities. For instance, when modeling something like rolling a die, whether we explicitly define it or just use a model that others have already established, we still need the sample space, the set of events, and the probability function. Regardless of whether we know how to use these elements explicitly or not, they are essential components.

It is indeed a fact. So, up to this point, you can understand that there is something called probability theory, and that measuring the size of a set is fundamental to probability. Just having this basic understanding should suffice. Now, what are some examples? For instance, consider an interval where we have xn - 1 and xn, starting from 0 and going up to 1. Therefore, the size of the entire set is 1. Now, if we take the probability of picking an element from that interval, we can say we are looking at xn - xn - 1. That seems fine, right? However, this leads to a situation where it intersects with the segments. When that happens, what occurs? It leaves just a single point. If it ends up leaving just one point, what happens? The length of the interval becomes 0. So, I picked xn, but the probability of picking this number is 0, which creates a peculiar situation.

I don't think you need to take it too seriously. First of all, in situations like this, what I want to say is that when we create a mathematical model of the real world, it doesn't always have to perfectly describe reality or align with our intuition. There can be instances that contradict our intuition. For example, the idea that an event with a probability of 0 cannot occur is merely a human interpretation; mathematically, it has been defined that way, and there is no problem with the concept of 0 itself. Personally, I find this explanation a bit better, but a more general explanation would be as follows: Let's think about it again. We have numbers from 0 to 1, and if there is a position at 1, and another position at 1, when we pick a number uniformly between them...

The probability of picking a number in between is 1. However, if there is an interval here, and the start of the interval is 1 and the end is also 1, then the length of the interval is 0. So what happens? The interval from 1 to 13 actually consists of just that one point. So, what is the probability of picking 13? Since we said the length of the interval represents probability, and the length of the interval is 0, it means the probability of picking 1 is 0. The question arises: how can it be 0 if I actually picked it? This brings us to the core of the paradox. My first explanation is that whether we understand it realistically or not, mathematically, it is constructed this way and it doesn't really matter. But the mathematics operates that way, regardless of our understanding. My second explanation is this: typically...

Generally, when people explain this, they say that when we measure our weight, for example, if my weight is 68, it doesn't mean we are talking about 68.00. Instead, when we say 68, we are referring to a range, like between 67.5 and 68.5. So, when we mention a number, we are not talking about a number with infinite precision. We always refer to a range. For instance, if I say I picked 1, that is an overly idealistic situation. In reality, if there is a 1, we are looking at it as if we picked a number within a certain range around 1. Therefore, we are not using infinite precision, and from that perspective, the numbers we discuss, especially those in the real world, should be understood as ranges rather than numbers with infinite precision.

If you look at it that way, it's not about your body weight being this way; it's always talked about in this manner. When we say 68, that's actually what we're referring to. If there were a more precise scale, it might show 67.95 or 68 point-something. Anyway, something like that exists, right? So, we have to interpret it that way. You could also interpret it in another way. There is a slight conflict with our intuition regarding probabilities, but it's not something to worry too much about. We can interpret it in one way or in another way. Mathematics is inherently like that; you need to maintain a good balance. So, there is a real world, and when mathematics exists, you need to interpret this real world mathematically and bring the results back from there. However, in the intermediate part, you need a certain degree of flexibility and imagination. This is what we call modeling. Well, sociologists, right? Sociology is...

The truth is that it's an application of psychology. It's an application of psychology and biology, and it directly relates to chemistry. All of that is, well, an application of physics. Now, when we look at it this way, we see it as all being applied sciences. However, in mathematics, you guys are asking if there were any such people. It seems like mathematics discusses something a bit detached from reality. Nevertheless, since we solve problems from this perspective, we need to understand and be aware of mathematics to some extent. So, when we say we are working on a probability model, we create a sample space, then create events, and for each event, we assign probabilities to each subset. This process becomes the foundation for building a probability model. Now that we've talked about models, let's delve a little deeper into them.

Let's talk a bit more about related topics. We usually say that the probability of getting heads when flipping a coin is 1, and we proceed with this discussion. However, this is really just a commonly used model by many people. If I wanted to create a more sophisticated model, how would I do that? Well, I could create a sample space that includes heads, tails, and sides. By including these elements, we can create a more comprehensive model. In this case, the events would include the empty set, and since we have three elements in total, the events would be represented by the powerset of these elements. The probability of heads coming up can be around 0.495 or 0.497, and for the sides, you can calculate it appropriately.

It can be 0, 0.008, right? Anyway, within the range that does not violate the probability axioms mentioned earlier, you can create such models. It is open. However, the models we generally use are not like this; they are typically 1. But if you find that the probability of an event deviates significantly from 1 due to some asymmetry, you can design a new model. There are fields for this. For example, in estimation, we do such things. Since I don't know how much it will yield, I can set it as alpha, beta, and gamma, with the sum of alpha, beta, and gamma being 1. Each of these is greater than or equal to 0, and we can perform estimations on this. There may be fields of estimation as well.

There are many disciplines that are based on probability, and statistics is one of them. One of the important questions in statistics is whether something has an effect or not. Of course, there are various other questions, but generally, we often ask whether a new treatment is effective, whether a particular method works, or whether a person has the ability to do something, or if they just happened to get it right by chance. These are the kinds of questions that arise. Therefore, a crucial aspect of statistics is hypothesis testing. For example, although you may not hear much about it these days, when I was younger, there were opportunities for people to become chick sexers, where they would be recruited to determine the sex of chicks.

There was an academy, and if you watch the movie Mii, it features a profession that involves chick sexing, which is the ability to distinguish between male and female chicks. In the film, a person was able to differentiate ten chicks, getting eight of them correct. This raises the question of whether this person has the ability to distinguish or not. This judgment of ability can be seen as a divine realm, while what humans do follows a certain logical structure. 

What kind of logical structure is it? First, there is H0, known as the null hypothesis. In English, it is referred to as the null hypothesis, which sounds grand, but ultimately, it means that there is no hypothesis. What does this mean? If I have no additional information, I will just consider this as the default. So, even if someone claims they have the ability to sex chicks, I will...

Unless there is evidence to the contrary, one might think that there is nothing, which is the null hypothesis. For example, in the case of the alternative hypothesis, it would be the ability to distinguish. So, how do we determine if this is H0 or H1? The logical structure for this judgment is very simple. We assess the probability of observing the current data under the assumption that H0 is true. In other words, we calculate the probability of observing what I have seen, or something even more extreme, under the assumption that the null hypothesis is true. Therefore, under the assumption that you do not have the ability to distinguish between male and female chicks, we calculate the probability of getting eight correct answers, which is even more extreme than eight, nine, or ten. The probability of getting eight or more correct answers is very low. So, how should we interpret this?

Under the null hypothesis, a very low probability was observed. So, we reject the null hypothesis and accept H1. When we calculated it, the probability turned out to be quite high—around 30%. Well, a 30% probability could certainly occur; if you flip a coin ten times, you could get three heads. If the numbers come out too large, we maintain the null hypothesis. This is the logical structure. Just by listening to this, you can understand that in statistics, the concept of probability is actually foundational, and you will realize the importance of understanding probability. Therefore, if there is no additional increase, we must believe certain things. For example, if someone brings a new drug claiming it is effective, until he proves it, the null hypothesis would be that it is not effective. Under the null hypothesis, which states that it is not effective or that it performs worse or the same as existing drugs, we examine cases of treatment in clinical settings.

The results of the experiments show that if the probability of obtaining the current clinical trial results is somewhat high, then the null hypothesis is correct. So, it could certainly be possible. However, if the probability of observing the current values under the null hypothesis is extremely low, then we can argue that it does not adequately explain the results, leading us to reject the null hypothesis and accept the alternative hypothesis. 

Now, regarding how humans perform this task—whether a person can truly distinguish between chicks, for example—it's fundamentally about the assertion that if what you said is not true, then the probability of such a low occurrence is significant. So, we consider it to be true to some extent. The question then is, what is the boundary for this? The boundary is essentially up to the user’s discretion, or rather, the perspective of the one funding the research. Typically, in academia, a commonly used threshold is...

It is 0.5, which means that if something occurs that is less than 5%, well, it's a bit hard to explain. If a number greater than 0.1 comes up, that could definitely happen. In the case of the graph, it expresses that. Some of you may be writing a paper and working hard to create something, but when you assess its effectiveness, the value you mentioned earlier turns out to be 0.049. So what happens? It's less than 0.05, right? Therefore, you reject H0 and accept H1, meaning that the current research result survives. If this number had been 0.051, then it would be dismissed, concluding that the research result has no effect. So, there is a certain significance at the 0.05 level. However, aside from this, there are things that people can do.

I think it's also true that there is no other way to determine this. So, how does a person judge whether a drug is effective or not? First of all, saying that a drug is not effective means that it is not better or worse compared to existing drugs. Based on that premise, we calculate the probability of observing the data we currently have. If the probability of observing the current data is very low, we conclude that the drug is effective. If the probability is moderately high, we might say, Well, this could just be a coincidence, and conclude that it is not effective. So, the boundary is at 0.05. If a 0.05 result comes out, it might feel a bit uneasy, but we should celebrate with champagne. So, we have looked at how probability is applied and how you should create models.

The size of the intersection of A and B can be thought of intuitively. The size of the intersection of A and B is the size of A plus the size of B minus the size of the union. We could express it this way: the size of A union B can also be represented like this. The concept of probability fundamentally measures the size of a set, which is very important. This brings us to conditional probability. We have talked extensively about the probability of A, but what we are really interested in going forward is what happens when a condition is given. In this case, it is B that needs to be considered. What will mostly go in the B position in the future are observational values. The value that goes in that position is what I am curious about. So, if I am curious about a certain value and it is observable, then that value can be determined. However, the situation becomes complicated when we have to think about probabilities in more complex scenarios.

This was observed as a cause or underlying factor. What I am curious about is that I saw it with my own eyes, and I am drawing it with my eyes. This is a common situation, and in a way, a significant portion of the problems we humans need to solve will be expressed in such a manner. The relationship is that I have made an observation, and based on that, I believe there is a probability of event B occurring. So, to go in that direction, conditional probability is defined like this. Understood? For this to be defined, B must not be zero, and if the intersection of A and B is an empty set, then the occurrence of B implies that A did not occur. This would be a zero probability. Here, we are just practicing. So, for example, if we have a resistance of 5% tolerance, it means that resistance could be, let's say, 10 ohms, or...

We talk about 47 ohms and 100 ohms, but the 5% tolerance means that it’s not exactly ohms; it could be 95 or 105. So, even resistors, from this perspective, can be modeled as random variable modeling. You could model that as a stunt, or you could solve it using random variable modeling. One way to control or handle these uncertainties is to think of them as random variables. Anyway, if we draw 47, there are four of them, so that’s 44 out of 4. Drawing a resistor with a 5% tolerance means there are two 60 ohm resistors, which gives us 62 out of 6. For the 100 ohm resistors, we have 32, so that’s 32 out of 100. Now, we have no specific conditions; now it will be conditional, an intersection of A and B'.

It was 47 ohms, and since there is 5% of B, there are only 28. There obviously won't be 100 at 47 ohms. If you look at it like this, you would naturally know that if I know it's ohms, it can't be 47 ohms; it becomes zero. In this case, if I know 5%, then 47 would become 22 out of 28. This will be conditional probability. What comes out here is called the total probability formula. This refers to the overall probability. The total probability formula is very intuitive and is quite useful for solving difficult problems. First of all, it's good to know the term partition. When we say we partition a whole set S, it’s like dividing land where all the owners exist and dividing the land. So, B1 is someone's land.

When it comes to B2, when someone's land becomes like this, the whole thing can be expressed as a union of the B2s. It's the union of the whole S, and additionally, there is no intersection with anyone. So, it means that some area is well-divided among the dogs, and there is no joint ownership. You can think of it that way. This is now S partitioned. When we say we partitioned S, and when we say we partitioned the sample space B, this is a method to calculate probability. Now, this is the formula related to the partition, the condition for calculating A. What methods are available? Well, it could come from the axioms of probability, but we can write it as PA, PA intersect B1, PA intersect B2, and so on, up to PA intersect BN. This is possible because they are all mutually disjoint. Since B1, B2, and B3 are disjoint, each one is disjoint.

It will be disjoint, but since the total is included, this holds true based on the third axiom we discussed. If we write each of these in conditional probability form, it will look something like this: A1 should be excluded. Now, you might think that the left side is easier, and wonder why it is written so complicatedly. However, as you will see later, trying to think about what we know about A all at once can be quite difficult. But when we add conditions—what happens under condition A, what happens under condition B—it becomes something we can tackle. So, in a situation where we cannot solve this large chunk all at once, we break it down to solve each part separately, which is essentially the formula for total probability. Therefore, it becomes a very useful formula.

[05-02] 확률 Crash Course

[05-02] 확률 Crash Course