Pooling is an aggregation operation that transforms a set of input vectors (or a grid of numerical features) into a single representative vector – the so-called pooling vector. It captures the most essential information from the input while significantly reducing its dimensionality. Think of it as an "executive summary" of a data group – it keeps the core message but discards unnecessary fine-grained details.

For instance, imagine you have four students, each represented by a vector of 5 test scores. The pooling vector would be a single student with 5 scores that summarize the performance of the entire group. In computer vision, pooling shrinks image feature maps; in natural language processing, it turns a sequence of word embeddings into one compact semantic vector for the whole sentence.

The Gaussian distribution (also known as the Normal distribution) is the most fundamental probability distribution in statistics and machine learning. Its characteristic bell-shaped curve appears everywhere — from measurement errors in physics to heights and test scores in biology and education.

A single Gaussian is defined by two parameters: its mean ($\mu$), which sets the center of the curve, and its variance ($\sigma^2$), which determines the spread (width). However, real-world data is often multi-modal — it clusters around several distinct centers. This is where the Gaussian Mixture Model (GMM) comes in: it combines multiple Gaussian components, each with its own mean and variance, to model complex, multi-modal distributions.

Bayes' Theorem is a fundamental rule in probability theory that describes how to update the probability of a hypothesis as more evidence or information becomes available. It provides a mathematical connection between a prior belief and observed data, producing an updated posterior belief.

In simple terms, Bayes' theorem gives us a rigorous mathematical framework for learning from evidence.

Imagine that a patient is tested for a rare disease. Suppose the disease affects 1 in 10,000 people, the test has 99% sensitivity, and 99% specificity. What is the probability that a person who tests positive actually has the disease?

It is tempting to answer "99%", but that would be incorrect. The answer depends not only on the performance of the test, but also on the base rate — how common the disease is in the population.

If we test 10,000 people, approximately 1 person has the disease. That person has a 99% chance of testing positive, giving about 0.99 true positives.

Among the 9,999 healthy people, a 1% false-positive rate produces about 99.99 false positives.

Therefore, among all positive results, only a small fraction correspond to people who actually have the disease:

\[ P(\text{Disease} \mid \text{Positive}) = \frac{ P(\text{Positive} \mid \text{Disease})P(\text{Disease}) }{ P(\text{Positive}) } \approx 0.0098 \]

So the probability that a person who tests positive actually has the disease is only about 0.98%.

This is a classic illustration of the importance of the base rate. It also demonstrates why we must distinguish between:

\[ P(\text{Positive} \mid \text{Disease}) \]

and:

\[ P(\text{Disease} \mid \text{Positive}) \]

These are different probabilities. Bayes' theorem provides the mathematical connection between them.

A covariance matrix (also called the variance-covariance matrix) is the natural extension of covariance when we have more than two variables. While ordinary covariance measures the direction of the relationship between a single pair, the covariance matrix organises all pairwise covariances into one elegant, square structure. It is the fundamental building block of multivariate statistics, machine learning, and modern finance.

Imagine you are studying a dataset of houses. Instead of just looking at size and price, you also include number of bedrooms and age of the property. Now you have 4 variables, which means 4×4 = 16 possible relationships (including each variable with itself). The covariance matrix arranges all of them in a single table – like a "relationship map" of your entire dataset.