Understanding Probability Theory for Machine Learning Applications

Probability theory forms the foundation for understanding machine learning, emphasizing the importance of modeling and axioms.

### Understanding Probability in Machine Learning
The crash course aims to grasp the **essential aspects of probability for machine learning**, focusing on the **big picture** rather than a strong mathematical foundation. Probability is defined through a sample space and its subsets, with a probability function assigning numbers to events based on specific rules, known as **axioms**. These axioms establish that the size of an event's probability must be **between 0 and 1**, the total size of the sample space must be **1**, and the size of the union of non-intersecting sets equals the **sum of their sizes**.

### The Importance of Sample Space, Events, and Probabilities in Modeling
A proper model for a natural phenomenon requires a **sample space**, a set of **events**, and **probabilities**. The paradox of picking a number from an interval highlights that an event with a **probability of 0** can still occur, challenging our intuition. Real-world numbers should be understood as **ranges** rather than having infinite precision, as we often refer to a range when discussing measurements.

### The Role of Mathematics and Probability in Understanding the Real World
Mathematics requires a balance between **interpreting the real world** and applying **flexibility and imagination** through modeling. Probability models are created by defining a **sample space**, events, and assigning **probabilities to each subset**. Statistics, as a discipline based on probability, often focuses on **hypothesis testing** to determine the **effectiveness of treatments or methods**.

### The Logical Structure of Hypothesis Testing
The movie Mii features chick sexing, raising the question of a person's ability to **distinguish between male and female chicks**. The logical structure for judging this ability involves the **null hypothesis (H0)** and the **alternative hypothesis (H1)**, with probability calculations used to determine which hypothesis to accept. A **low probability of observing the current data under H0** leads to its rejection and the acceptance of H1, highlighting the **foundational role of probability in statistical analysis**.

### Understanding Probability and Its Application
Probability values play a crucial role in determining the **significance of research results**, with a threshold of **0.05** being common for rejecting the null hypothesis. The effectiveness of a drug is assessed by calculating the probability of observing the current data, with **low probabilities indicating effectiveness**. Conditional probability is important for understanding the relationship between events, and it is defined based on the **occurrence of one event given the condition of another**.

### Modeling Resistor Tolerances and Probability
Resistors with **5% tolerance** can vary in value, making them suitable for **random variable modeling**. The total probability formula is used to calculate overall probabilities by **partitioning a set** and considering **mutually disjoint events**. This approach helps in tackling complex problems by **breaking them down into smaller, manageable parts**.

### Understanding Probability: Foundations for Machine Learning and Statistical Modeling
* 📚 Our primary goal is to grasp the probabilities that stem from machine learning and deep learning through a crash course focusing on **essential probability concepts** rather than deep mathematical foundations.
* 🔍 Probability is mathematically defined as the **likelihood of an event occurring**, with a sample space that contains measurable subsets, such as sizes of **0.3, 0.2, and 0.4**.
* 📏 The probability function assigns numbers to events, adhering to three axioms: the **size must be between 0 and 1**, the **total size of the set must equal 1**, and the **size of the union of non-overlapping sets equals the sum of their sizes**.
* 📊 The size of the empty set is proven to be **0** using these axioms, reinforcing that probability theory fundamentally measures the **size of sets** and is crucial for **modeling real-world phenomena**.
* 🎲 When creating probability models, we must define a sample space, events, and assign probabilities, which can lead to **intuitive paradoxes**, such as the probability of selecting a **specific point from a continuous interval being 0**.
* 📈 In statistics, hypothesis testing is a key application of probability, where we **assess the likelihood of observing data under a null hypothesis (H0)** to determine the **effectiveness of treatments or abilities**.
* 🔬 A common threshold for rejecting H0 is **0.05**, indicating that if the probability of observing the data under H0 is **low**, we may accept the alternative hypothesis (H1) as **valid**.
* 📐 Conditional probability and the total probability formula become **essential tools** in tackling complex probability scenarios, allowing us to **partition sample spaces** and **calculate probabilities effectively**.

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[05-02] 확률 Crash Course

[05-02] 확률 Crash Course