Even though most of us are introduced to the subject through this example, fitting functions to a training dataset seemingly doesn't give us any deep insight about the data.
This is what's working behind the scenes!
🧵 👇🏽
Consider a simple example: predicting the value 𝑦 from the observation 𝑥; for instance 𝑦-s are real estate prices based on the square footage 𝑥.
If you are a visual person, this is how you can imagine such dataset.
The first thing one would do is to fit a linear function 𝑓(𝑥) = 𝑎𝑥 + 𝑏 on the data.
By looking at the result, we can see that something is not right. Sure, it might capture the mean value for a given observation, but the variance and the noise in the data is not explained.
Next, we might try to fit a more expressive function, say a polynomial, but that only seems to make things worse by potentially overfitting on the training dataset.
We need an entirely different model to really explain the dataset.
This is where probabilities come in.
Instead of a deterministic function, we estimate the probability distribution of the observations.
If 𝑋 is the distribution of our data and 𝑌 is the corresponding observation, we can model their relation with a Gaussian distribution.
We can fit this model by maximizing the likelihood function.
Essentially, for a given set of parameters 𝑎 and 𝑏, the likelihood describes the probability that we observe the training data.
The higher it is, the better the model fits.
It turns out that maximizing the likelihood is the same as minimizing the Mean Squared Loss!
(Don't worry about the computational details yet, I'll have you covered soon.)
The result?
A probabilistic model that explains the entire dataset, not just its mean. Every time you fit a linear regressor, probability and statistics are working in the background.
Of course, there is much more behind the surface.
If you are interested in the technical details, check out my post below!
(All mathematical prerequisites on probability are covered.)
The Law of Large Numbers is one of the most frequently misunderstood concepts of probability and statistics.
Just because you lost ten blackjack games in a row, it doesn’t mean that you’ll be more likely to be lucky next time.
What is the law of large numbers, then? Read on:
The strength of probability theory lies in its ability to translate complex random phenomena into coin tosses, dice rolls, and other simple experiments.
So, let’s stick with coin tossing.
What will the average number of heads be if we toss a coin, say, a thousand times?
To mathematically formalize this question, we’ll need random variables.
Tossing a fair coin is described by the Bernoulli distribution, so let X₁, X₂, … be such independent and identically distributed random variables.
Matrix factorizations are the pinnacle results of linear algebra.
From theory to applications, they are behind many theorems, algorithms, and methods. However, it is easy to get lost in the vast jungle of decompositions.
This is how to make sense of them.
We are going to study three matrix factorizations:
1. the LU decomposition, 2. the QR decomposition, 3. and the Singular Value Decomposition (SVD).
First, we'll take a look at LU.
1. The LU decomposition.
Let's start at the very beginning: linear equation systems.
Linear equations are surprisingly effective in modeling real-life phenomena: economic processes, biochemical systems, etc.