Tivadar Danka Profile picture
Dec 27, 2021 15 tweets 4 min read Read on X
Entropy is not the easiest thing to understand.

It is rumored to describe something about information and disorder, but it is unclear why.

What do logarithms and sums have to do with the concept of information?

Let me explain!

↓ A thread. ↓ Image
I have randomly selected an integer between 0 and 31.

Can you guess which one? You can ask as many questions as you want.

What is the minimum number of questions you have to ask to be 100% sure?

You can start guessing the numbers one by one, sure. But there is a better way!
If you ask, "is the number larger or equal than 16?" you immediately eliminate half the search space!

Continuing with this tactic, you can find the number for sure in 5 questions.
In other words, we need to take the base two logarithm of 32 to get the number of questions required.

This logic applies to all numbers! If I pick a number between 0 and 𝑛-1, you need 𝑙𝑜𝑔(2, 𝑛) questions to find it for sure, by cutting the possibilities in half with each.
Because the answers are yes-or-no questions, we can encode each with a 0 or 1.

If we write down the answers in a row, we effectively encode the numbers in 𝑛 bits!

𝟎: 00000
𝟏: 00001
𝟐: 00010
...
𝟑𝟏: 11111

Each "code" is simply the number in base 2!
No matter which number I pick, five questions are needed to find it.

So, the average number of bits needed is also five.

However, we use a critical assumption here: I pick each number with an equal probability.

What if that is not the case?
Let's say I am picking between 0, 1, and 2, but I am picking 0 at 50% of the time, while 1 and 2 only 25% of the time.

We should put this into mathematical form!

Let's denote the number I pick with 𝑋. This is a random variable.

How many bits do we need now? Image
We can be more bit-efficient than before! Consider this.

1st question: did you pick 0?
If the answer is yes, the 2nd question is not needed. If not, we proceed!

2nd question: did you pick 1?
No matter what the answer is, we know the solution! Yes implies 1, no implies 2.
Following this idea, we can calculate the average number of bits as below. Image
(This is just the expected value of the number of bits.

If you didn't understand this step, check out my explanation about the expected value!)

)
Now we are almost there! Let's see the general case.

Suppose I pick between 𝑥₁, 𝑥₂, ..., 𝑥ₙ, and I pick 𝑥ₖ with probability 𝑝ₖ.

As before, the number of questions needed to find 𝑘 is the base two logarithm of 1/𝑝ₖ! Image
So, the entropy of a random variable is simply the average bits of information needed to guess its value successfully! Even though the formula is complicated, its meaning is simple.

Entropy is simpler than you thought! (And probably also simpler than what you were taught.) Image
Having a deep understanding of math will make you a better engineer. I want to help you with this, so I am writing a comprehensive book about the subject.

If you are interested in the details and beauties of mathematics, check out the early access!

tivadardanka.com/book
A few extra comments!

1. What happens if the logarithm of the probability is not an integer?

Not all questions provide 100% new information. Sometimes, the answer is partially contained in other bits.

Hence, the "amount of new information" is not always an integer.
2. Does the base of the logarithm matter?

In general, we can easily swap the base of the logarithms, as shown below.

Thus, swapping bases in the entropy formula is just multiplication with a constant. Image

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More from @TivadarDanka

Jul 14
"Probability is the logic of science."

There is a deep truth behind this conventional wisdom: probability is the mathematical extension of logic, augmenting our reasoning toolkit with the concept of uncertainty.

In-depth exploration of probabilistic thinking incoming. Image
Our journey ahead has three stops:

1. an introduction to mathematical logic,
2. a touch of elementary set theory,
3. and finally, understanding probabilistic thinking.

First things first: mathematical logic.
In logic, we work with propositions.

A proposition is a statement that is either true or false, like
• "it's raining outside",
• or "the sidewalk is wet".

These are often abbreviated as variables, such as A = "it's raining outside".
Read 28 tweets
Jul 13
Conditional probability is the single most important concept in statistics.

Why? Because without accounting for prior information, predictive models are useless.

Here is what conditional probability is, and why it is essential. Image
Conditional probability allows us to update our models by incorporating new observations.

By definition, P(B | A) describes the probability of an event B, given that A has occurred. Image
Here is an example. Suppose that among 100 emails, 30 are spam.

Based only on this information, if we inspect a random email, our best guess is a 30% chance of it being a spam.

This is not good enough. Image
Read 10 tweets
Jul 11
Most people think math is just numbers.

But after 20 years with it, I see it more like a mirror.

Here are 10 surprising lessons math taught me about life, work, and thinking clearly: Image
1. Breaking the rules is often the best course of action.

We have set theory because Bertrand Russell broke the notion that “sets are just collections of things.”
2. You have to understand the rules to successfully break them.

Miles Davis said, “Once is a mistake, twice is jazz.”

Mistakes are easy to make. Jazz is hard.
Read 12 tweets
Jul 8
This will surprise you: sine and cosine are orthogonal to each other.

What does orthogonality even mean for functions? In this thread, we'll use the superpower of abstraction to go far beyond our intuition.

We'll also revolutionize science on the way. Image
Our journey ahead has three milestones. We'll

1. generalize the concept of a vector,
2. show what angles really are,
3. and see what functions have to do with all this.

Here we go!
Let's start with vectors. On the plane, vectors are simply arrows.

The concept of angle is intuitive as well. According to Wikipedia, an angle “is the figure formed by two rays”.

How can we define this for functions? Image
Read 18 tweets
Jul 7
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
Jul 5
If I had to learn Math for Machine Learning from scratch, this is the roadmap I would follow: Image
1. Linear Algebra

These are non-negotiables:

• Vectors
• Matrices
• Equations
• Factorizations
• Matrices and graphs
• Linear transformations
• Eigenvalues and eigenvectors

Now you've learned how to represent and transform data. Image
2. Calculus

Don't skip any of these:

• Series
• Functions
• Sequences
• Integration
• Optimization
• Differentiation
• Limits and continuity

Now you understand the math behind algorithms like gradient descent and get a better feeling of what optimization is. Image
Read 6 tweets

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