Tivadar Danka Profile picture
Aug 23, 2021 9 tweets 3 min read Read on X
Machine learning is more than function fitting.

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.)

tivadardanka.com/blog/the-stati…

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More from @TivadarDanka

Oct 14
In machine learning, we use the dot product every day.

However, its definition is far from revealing. For instance, what does it have to do with similarity?

There is a beautiful geometric explanation behind: Image
By definition, the dot product (or inner product) of two vectors is defined by the sum of coordinate products. Image
To peek behind the curtain, there are three key properties that we have to understand.

First, the dot product is linear in both variables. This property is called bilinearity. Image
Read 15 tweets
Oct 11
Behold one of the mightiest tools in mathematics: the camel principle.

I am dead serious. Deep down, this tiny rule is the cog in many methods. Ones that you use every day.

Here is what it is, how it works, and why it is essential: Image
First, the story:

The old Arab passes away, leaving half of his fortune to his eldest son, third to his middle son, and ninth to his smallest.

Upon opening the stable, they realize that the old man had 17 camels. Image
This is a problem, as they cannot split 17 camels into 1/2, 1/3, and 1/9 without cutting some in half.

So, they turn to the wise neighbor for advice. Image
Read 18 tweets
Oct 9
Matrix multiplication is not easy to understand.

Even looking at the definition used to make me sweat, let alone trying to comprehend the pattern. Yet, there is a stunningly simple explanation behind it.

Let's pull back the curtain! Image
First, the raw definition.

This is how the product of A and B is given. Not the easiest (or most pleasant) to look at.

We are going to unwrap this. Image
Here is a quick visualization before the technical details.

The element in the i-th row and j-th column of AB is the dot product of A's i-th row and B's j-th column. Image
Read 16 tweets
Oct 8
Graph theory will seriously enhance your engineering skills.

Here's why you must be familiar with graphs: Image
What do the internet, your brain, the entire list of people you’ve ever met, and the city you live in have in common?

These are all radically different concepts, but they share a common trait.

They are all networks that establish relationships between objects. Image
As distinct as these things seem to be, they share common properties.

For example, the meaning of “distance” is different for

• Social networks
• Physical networks
• Information networks

But in all cases, there is a sense in which some objects are “close” or “far”. Image
Read 14 tweets
Oct 7
One of the coolest ideas in mathematics is the estimation of a shape's area by throwing random points at it.

Don't believe this works? Check out the animation below, where I show the method on the unit circle. (Whose area equals to π.)

Here is what's behind the magic:
Let's make this method precise!

The first step is to enclose our shape S in a square.

You can imagine this as a rectangular dartboard. Image
Now, we select random points from the board and count how many hit the target.

Again, you can imagine this as closing your eyes, doing a 360° spin, then launching a dart.

(Suppose that you always hit the board. Yes, I know. But in math, reality doesn't limit imagination.) Image
Read 14 tweets
Oct 6
The way you think about the exponential function is wrong.

Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?

Here is what's really behind the most important function of all time: Image
First things first: terminologies.

The expression aᵇ is read "a raised to the power of b."

(Or a to the b in short.) Image
The number a is called the base, and b is called the exponent.

Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.

Sounds simple enough. Image
Read 18 tweets

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