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.)
"1. No income tax for women with at least two children for life."
This is an election hack, meant to buy votes for the upcoming 2026 election. Fidesz (Hungary's ruling party) is significantly down in the polls after it was leaked that a convicted p*d*ph*le accessory was given a presidential pardon.
Hell, they even let a child p*rn*gr*phy wholesaler with 96000 images on his computer walk away with ~$1500 fine. (Check en.wikipedia.org/wiki/G%C3%A1bo… if you don't believe me.)
Thus, the government is scraping to buy back the trust of families.
Even if it wasn't an empty promise, waiving the income tax is unrealistic for budgetary reasons. Hungary's economy is in the toilet.
"3. Housing incentives for young couples.
Offers a low interest loan for couples raising or committing to having one child or more."
This loan is another propaganda trick. In practice, this loan resulted in the biggest housing crisis of the country's history, because all it did was raise the price of every real estate by the amount of the loan, making real estate ownership virtually impossible for the young generation.
No matter the field, you can (almost always) find a small set of mind-numbingly simple ideas making the entire thing work.
In machine learning, the maximum likelihood estimation is one of those.
I'll start with a simple example to illustrate a simple idea.
Pick up a coin and toss it a few times, recording each outcome. The question is, once more, simple: what's the probability of heads?
We can't just immediately assume p = 1/2, that is, a fair coin.
For instance, one side of our coin can be coated with lead, resulting in a bias. To find out, let's perform some statistics! (Rolling up my sleeves, throwing down my gloves.)
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?
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.
The expected value is one of the most important concepts in probability and statistics.
For instance, all the popular loss functions in machine learning, like cross-entropy, are expected values. However, its definition is far from intuitive.
Here is what's behind the scenes.
It's better to start with an example.
So, let's play a simple game! The rules: I’ll toss a coin, and if it comes up heads, you win $1. However, if it is tails, you lose $2.
Should you even play this game with me? We’ll find out.
After n rounds, your earnings can be calculated by the number of heads times $1 minus the number of tails times $2.
If we divide total earnings by n, we obtain your average earnings per round.
The single biggest argument about statistics: is probability frequentist or Bayesian?
It's neither, and I'll explain why.
Buckle up. Deep-dive explanation incoming.
First, let's look at what is probability.
Probability quantitatively measures the likelihood of events, like rolling six with a dice. It's a number between zero and one. This is independent of interpretation; it’s a rule set in stone.
In the language of probability theory, the events are formalized by sets within an event space.
The event space is also a set, usually denoted by Ω.)