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Tivadar Danka Profile picture
@TivadarDanka
Apr 27, 2021 • 6 tweets • 2 min read • Read on X
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Have you ever wondered why include the logarithm in the definition of log-likelihood?

The answer is simple: logarithm makes differentiation of products easier.

Let's see why!

🧵 👇🏽
Although the derivative of a sum is the sum of derivatives, a similar property cannot be stated about the product of functions.

The derivative of a product is slightly more complicated: it is a sum of products.
The formula gets even more complicated when we have more functions in the product.

When potentially hundreds of terms are present, like in the likelihood function, computing this is not feasible.
This is where taking the logarithm can help us: it turns multiplication into addition!
So, instead of differentiating a product of 𝑛 functions, we have to deal with the sum!

Since the logarithm strictly increasing, it won't interfere with optimization either: the optimums are attained at the same place.
If you enjoyed this explanation, consider following me and hitting a like/retweet on the first tweet of the thread!

I regularly post simple explanations of seemingly complicated concepts in machine learning, make sure you don't miss out on the next one!

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More from @TivadarDanka

Tivadar Danka Profile picture
Oct 25
The following multiplication method makes everybody wish they had been taught math like this in school.

It's not just a cute visual tool: it illuminates how and why long multiplication works.

Here is the full story: Image
First, the method.

The first operand (21 in our case) is represented by two groups of lines: two lines in the first (1st digit), and one in the second (2nd digit).

One group for each digit.
Similarly, the second operand (32) is encoded with two groups of lines, one for each digit.

These lines are perpendicular to the previous ones.
Read 10 tweets
Tivadar Danka Profile picture
Oct 21
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
Tivadar Danka Profile picture
Oct 20
In calculus, going from a single variable to millions of variables is hard.

Understanding the three main types of functions helps make sense of multivariable calculus.

Surprisingly, they share a deep connection. Let's see why: Image
In general, a function assigns elements of one set to another.

This is too abstract for most engineering applications. Let's zoom in a little! Image
As our measurements are often real numbers, we prefer functions that operate on real vectors or scalars.

There are three categories:

1. vector-scalar,
2. vector-vector,
3. and scalar-vector. Image
Read 16 tweets
Tivadar Danka Profile picture
Oct 19
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: Image
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. Image
Read 17 tweets
Tivadar Danka Profile picture
Oct 15
I have spent at least 50% of my life studying, practicing, and teaching mathematics.

The most common misconceptions I encounter:

• Mathematics is useless
• You must be good with numbers
• You must be talented to do math

These are all wrong. Here's what math is really about: Image
Let's start with a story.

There’s a reason why the best ideas come during showers or walks. They allow the mind to wander freely, unchained from the restraints of focus.

One particular example is graph theory, born from the regular daily walks of the legendary Leonhard Euler.
Here is the map of Königsberg (now known as Kaliningrad, Russia), where these famous walks took place.

This part of the city is interrupted by several rivers and bridges.

(I cheated a little and drew the bridges that were there in Euler's time, but not now). Image
Read 15 tweets
Tivadar Danka Profile picture
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

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