Entropy (for data science) Clearly Explained!!!

Summary

This StatQuest episode, hosted by Josh Starmer, dives into the concept of entropy for data science, explaining its applications in classification trees, mutual information, and algorithms like t-SNE and UMAP. The video illustrates how entropy quantifies surprise and similarity, using the example of chickens in different areas to demonstrate the relationship between probability and surprise. It then explores the calculation of surprise and entropy, highlighting the importance of entropy in measuring the expected surprise per event, such as coin flips, and its significance in data science.

Takeaways

• 📚 Entropy is a fundamental concept in data science used for building classification trees, mutual information, and in algorithms like t-SNE and UMAP.
• 🔍 Entropy helps quantify similarities and differences in data, which is crucial for various machine learning applications.
• 🤔 The concept of entropy is rooted in the idea of 'surprise', which is inversely related to the probability of an event.
• 🐔 The video uses the analogy of chickens of different colors in various areas to illustrate the relationship between probability and surprise.
• ⚖️ The level of surprise is not directly proportional to the inverse of probability due to the undefined log of zero when probability is zero.
• 📉 To calculate surprise, the logarithm of the inverse of the probability is used, which aligns with the concept of 'information gain'.
• 🎲 When flipping a biased coin, the surprise for getting heads or tails can be calculated using the log of the inverse of their respective probabilities.
• 🔢 Entropy is the expected value of surprise, calculated as the average surprise per event over many occurrences.
• ∑ The mathematical formula for entropy involves summing the product of the probability of each outcome and the surprise (logarithm of the inverse probability) of that outcome.
• 📈 Entropy can be represented in sigma notation, emphasizing its role as an expected value derived from the sum of individual probabilities and their associated surprises.
• 🌐 Entropy values can be used to compare the distribution of different categories within a dataset, with higher entropy indicating greater disorder or diversity.

Q & A

What is the main topic of the StatQuest video?

The main topic of the video is entropy in the context of data science, explaining how it is used to build classification trees, mutual information, and in algorithms like t-SNE and UMAP.

Why is understanding surprise important in the context of entropy?

Understanding surprise is important because it is inversely related to probability, which helps in quantifying how surprising an event is based on its likelihood of occurrence.

How does the video use chickens to illustrate the concept of surprise?

The video uses a scenario with two types of chickens, orange and blue, organized into different areas with varying probabilities of being picked, to demonstrate how the level of surprise correlates with the probability of an event.

Why can't we use the inverse of probability alone to calculate surprise?

Using the inverse of probability alone doesn't work because it doesn't correctly represent the surprise when the probability is very high, like when flipping a coin that always lands on heads.

What mathematical function is used to calculate surprise instead of just the inverse of probability?

The logarithm of the inverse of probability is used to calculate surprise, which gives a more accurate representation of the relationship between probability and surprise.

Why is the log base 2 used when calculating surprise for two outcomes?

The log base 2 is used for two outcomes because it is customary and it aligns with information theory principles, where entropy measures information in bits.

How does the video explain the concept of entropy in terms of flipping a coin?

The video explains entropy by calculating the average surprise per coin toss over many flips, which represents the expected surprise or entropy of the coin-flipping process.

What is the formula for entropy in terms of surprise and probability?

The formula for entropy is the sum of the product of each outcome's surprise and its probability, which can be represented using summation notation as the expected value of surprise.

How does the entropy value change with the distribution of chickens in different areas?

The entropy value changes based on the probability distribution of the chickens. Higher entropy indicates a more even distribution of chicken types, leading to a higher expected surprise per pick.

What is the significance of entropy in data science applications?

In data science, entropy is significant as it quantifies the uncertainty or surprise in a dataset, which is useful for building classification models, measuring mutual information, and in dimension reduction algorithms.

How does the video conclude the explanation of entropy?

The video concludes by demonstrating how entropy can be used to quantify the similarity or difference in the distribution of items, like chickens, and by providing a humorous note on surprising someone with the 'log of the inverse of the probability'.

Outlines

00:00

📊 Introduction to Entropy in Data Science

This paragraph introduces the concept of entropy in the context of data science, explaining its various applications such as in building classification trees and in algorithms like t-SNE and UMAP. It emphasizes that entropy is a measure of surprise, inversely related to probability, and is foundational in quantifying similarities and differences in data. The paragraph sets the stage for a deeper exploration of entropy by using the analogy of picking chickens of different colors from separate areas, illustrating how the level of surprise correlates with the probability of an event occurring.

05:01

🧐 Calculating Surprise and the Role of Probability

The second paragraph delves into the calculation of surprise, which is a precursor to understanding entropy. It discusses the relationship between probability and surprise, noting that surprise is highest when an event is least expected. The paragraph highlights the limitations of using the inverse of probability to calculate surprise, particularly when the probability is zero or one, and introduces the logarithmic approach to accurately represent the concept of surprise. It also explains how the surprise for a sequence of events is the sum of the individual surprises, providing a foundation for calculating entropy.

10:03

📉 Understanding Entropy as Average Surprise

This paragraph explains how entropy is derived from the concept of average surprise per event. It provides a step-by-step calculation of entropy using the example of a biased coin with varying probabilities of landing heads or tails. The paragraph illustrates how to calculate the total surprise for multiple events and then derives the entropy by averaging the surprise over the number of events. It also introduces the statistical notation for entropy as the expected value of surprise, emphasizing the cancellation of event counts in the calculation process.

15:04

🔄 Entropy as a Measure of Similarity and Difference

The final paragraph applies the concept of entropy to the original chicken analogy, calculating the entropy for different areas with varying ratios of orange and blue chickens. It demonstrates how entropy can be used to quantify the similarity or difference in the distribution of chickens, with higher entropy indicating a more even distribution and thus a higher expected surprise. The paragraph concludes with a humorous note on surprising someone with the 'log of the inverse of the probability' and a call to action for supporting the StatQuest channel through various means.

Keywords

💡Entropy

Entropy, in the context of this video, refers to a measure from information theory that quantifies the expected surprise in a random variable or a dataset. It is central to the theme as it helps explain how data can be classified and compared in terms of its disorder or unpredictability. For example, the video uses the concept of entropy to discuss its application in building classification trees and in algorithms like t-SNE and UMAP.

💡Data Science

Data Science is an interdisciplinary field that uses scientific methods, processes, algorithms, and systems to extract knowledge and insights from data. In the video, entropy is discussed as a fundamental concept in data science, particularly for tasks like building classification trees and quantifying relationships between data points.

💡Expected Values

Expected values are a fundamental concept in probability theory and statistics, representing the average or mean result of an experiment repeated over a large number of trials. The video assumes the viewer's familiarity with expected values as a prerequisite for understanding how entropy is calculated and used in data science.

💡Classification Trees

Classification Trees are a non-parametric supervised learning method used for classification and regression. They are mentioned in the video as one application of entropy, where entropy helps in deciding the best splits for the tree by measuring the impurity of the data at each node.

💡Mutual Information

Mutual Information is a measure used in statistics to quantify the amount of information obtained about one random variable through observing another random variable. The video explains that mutual information is based on entropy, and it helps in understanding the relationship between two variables.

💡Relative Entropy

Relative Entropy, also known as the Kullback-Leibler divergence, is a measure of how one probability distribution diverges from a second, expected probability distribution. The video mentions it as a basis for comparing two probability distributions, which is derived from the concept of entropy.

💡Cross Entropy

Cross Entropy is a measure of the difference in entropy between two probability distributions. In the video, it is mentioned as a concept that appears in various algorithms, including those for dimension reduction like t-SNE and UMAP, where it helps in measuring the difference between the distributions of data in high-dimensional and low-dimensional spaces.

💡Surprise

Surprise, in the context of this video, is a concept related to the unexpectedness of an outcome given its probability. It is inversely related to probability, where lower probabilities result in higher surprise. The video uses the concept of surprise to explain the calculation of entropy, emphasizing how it quantifies the unpredictability of data.

💡Logarithm

A logarithm is the inverse operation to exponentiation, used in mathematics and computer science to perform calculations with exponents. In the video, the logarithm of the inverse of probability is used to calculate surprise, which is then used to derive the concept of entropy.

💡Coin Flip

The coin flip is used as a simple example in the video to illustrate the concepts of probability, surprise, and entropy. It demonstrates how the expected surprise (entropy) can be calculated when a coin has a biased outcome, with heads appearing more frequently than tails.

💡Dimension Reduction

Dimension Reduction is a process in data analysis where the number of random variables under consideration is reduced, while still retaining most of the relevant information. The video mentions algorithms like t-SNE and UMAP, which use concepts derived from entropy, such as cross entropy, to perform dimension reduction effectively.

Highlights

Entropy is a concept used in data science for building classification trees and quantifying relationships.

Mutual information, relative entropy, and cross entropy are based on entropy and used in various algorithms including t-SNE and UMAP.

Entropy helps quantify similarities and differences in data.

Understanding surprise is fundamental to grasping entropy, as it is inversely related to probability.

The concept of surprise is illustrated through the example of picking chickens of different colors from various areas.

Calculating surprise involves using the log of the inverse of probability to avoid undefined values.

The log function is used to map the inverse probability to a scale where no surprise is zero.

Entropy is the average surprise per event, calculated by summing individual surprises and dividing by the number of events.

The entropy formula can be derived from the concept of expected surprise.

Entropy is represented mathematically as the expected value of surprise using sigma notation.

The standard form of entropy equation involves logarithms and probabilities, originally published by Claude Shannon.

Entropy can be used to measure the unpredictability or information content in a set of outcomes.

In the context of the chicken example, entropy quantifies the mix of orange and blue chickens in different areas.

Higher entropy indicates a more even distribution of outcomes, leading to greater surprise.

Entropy decreases as the difference in the number of two types of outcomes increases, indicating less surprise.

TheStatQuest channel offers study guides for offline review of statistics and machine learning.

Support for StatQuest can come in various forms including Patreon contributions, channel memberships, merchandise, or donations.

Transcripts

yes you can understand

entropy

hooray

steadquest

hello i'm josh starmer and welcome to

statquest today we're going to talk

about entropy for data science and it's

going to be clearly explained

note this stat quest assumes that you

are already familiar with the main ideas

of expected values if not check out the

quest

entropy is used for a lot of things in

data science

for example entropy can be used to build

classification trees

which are used to classify things

entropy is also the basis of something

called mutual information which

quantifies the relationship between two

things

and entropy is the basis of relative

entropy aka the colback liebler distance

and cross entropy

which show up all over the place

including fancy dimension reduction

algorithms like t snee and umap

what these three things have in common

is that they all use entropy or

something derived from it to quantify

similarities and differences

so let's learn how entropy quantifies

similarities and differences

however in order to talk about entropy

first we have to understand surprise

so let's talk about chickens

imagine we had two types of chickens

orange and blue and instead of just

letting them randomly roam all over the

screen

our friends stat squats chased them

around until they were organized into

three separate areas a b and c

now if statsquatch just randomly picked

up a chicken in area a

then because there are six orange

chickens and only one blue chicken there

is a higher probability that they will

pick up an orange chicken

and since there is a higher probability

of picking up an orange chicken it would

not be very surprising if they did

in contrast if statsquatch picked up the

blue chicken from area a we would be

relatively surprised

area b has a lot more blue chickens than

orange

and because there is now a higher

probability of picking up a blue chicken

we would not be very surprised if it

happened

and because there is a relatively low

probability of picking the orange

chicken

that would be relatively surprising

lastly area c has an equal number of

orange and blue chickens

thus regardless of what color chicken we

pick up we would be equally surprised

combined these areas tell us that

surprise is in some way inversely

related to probability

in other words when the probability of

picking up a blue chicken is low the

surprise is high

and when the probability of picking up a

blue chicken is high the surprise is low

bam

now we have a general intuition of how

probability is related to surprise

now let's talk about how to calculate

surprise

because we know there is a type of

inverse relationship between probability

and surprise

it's tempting to just use the inverse of

probability to calculate surprise

because when we plot the inverse we see

that the closer the probability is to

zero the larger the y-axis value

however there's at least one problem

with just using the inverse of the

probability to calculate surprise

to get a better sense of this problem

let's talk about the surprise associated

with flipping a coin

imagine we had a terrible coin and every

time we flipped it we got heads blah

blah blah blah

ugh flipping this coin is super boring

hey statsquatch how surprised would you

be if the next flip gave us heads

i would not be surprised at all

so

when the probability of getting heads is

one

then we want the surprise for getting

heads to be zero

however when we take the inverse of the

probability of getting heads we get one

instead of what we want

zero

and this is one reason why we can't just

use the inverse of the probability to

calculate surprise

so instead of just using the inverse of

the probability to calculate surprise

we use the log of the inverse of the

probability

now since the probability of getting

heads is 1 and thus we will always get

heads and it will never surprise us

the surprise for heads is zero

in contrast since the probability for

getting tails is zero and thus will

never get tails it doesn't make sense to

quantify the surprise of something that

will never happen

so when we plug in 0 for the probability

and use the properties of logs to turn

the division into subtraction

the second term is the log of 0

and because the log of 0 is undefined

the whole thing is undefined

and this result is ok because we're

talking about the surprise associated

with something that never happens

like the inverse of the probability the

log of the inverse of the probability

gives us a nice curve

and the closer the probability gets to

zero the more surprise we get

but now the curve says there is no

surprise when the probability is one

so surprise is the log of the inverse of

the probability

bam

note when calculating surprise for two

outputs in this case the two outputs are

heads and tails then it is customary to

use the log base 2 for the calculations

now that we know what surprise is

let's imagine that our coin gets heads

90 percent of the time

and it gets tails 10 percent of the time

now let's calculate the surprise for

getting heads

and tails

as expected because getting tails is

much rarer than getting heads the

surprise for tails is much larger

now let's flip this coin three times

and we get heads heads and tails

the probability of getting two heads and

one tail is

0.9 times 0.9 for the heads

times 0.1 for the tails

and if we want to know exactly how

surprising it is to get two heads and

one tail

then we can plug this probability into

the equation for surprise

and use the properties of logs to

convert the division into subtraction

and use the properties of logs to

convert the multiplication into addition

and then plug in chug and we get 3.62

but more importantly we see that the

total surprise for a sequence of coin

tosses is just the sum of the surprises

for each individual toss

in other words the surprise for getting

one heads is

0.15

and since we got two heads we add 0.15

two times

plus 3.32 for the one tail

to get the total surprise for getting

two heads and one tail

medium bam

now because this diagram takes up a lot

of space let's summarize the information

in a table

the first row in the table tells us the

probability of getting heads or tails

and the second row tells us the

associated surprise

now if we wanted to estimate the total

surprise after flipping the coin 100

times

we approximate how many times we will

get heads by multiplying the probability

we will get heads 0.9 by 100

and we estimate the total surprise from

getting heads by multiplying by 0.15

so this term represents how much

surprise we expect from getting heads in

100 coin flips

likewise we can approximate how many

times we will get tails by multiplying

the probability we will get tails 0.1 by

100

and we estimate the total surprise from

getting tails by multiplying by 3.32

so the second term represents how much

surprise we expect from getting tails in

100 coin flips

now we can add the two terms together to

find out the total surprise

and we get

46.7 hey statsquatch is back

ok

i see that we just estimated the

surprise for 100 coin flips

but aren't we supposed to be talking

about entropy

funny you should ask

if we divide everything by the number of

coin tosses 100

then we get the average amount of

surprise per coin toss 0.47

so on average we expect the surprise to

be 0.47

every time we flip the coin

and that is the entropy of the coin

the expected surprise every time we flip

the coin

double bam

in fancy statistics notation we say that

entropy is the expected value of the

surprise

anyway since we are multiplying each

probability by the number of coin tosses

100

and also dividing by the number of coin

tosses 100

then all of the values that represent

the number of coin tosses 100 cancel out

and we are left with the probability

that a surprise for heads will occur

times its surprise

plus the probability that a surprise for

tails will occur times its surprise

thus the entropy

0.47

represents the surprise we would expect

per coin toss if we flipped this coin a

bunch of times

and yes expecting surprise sounds silly

but it's not the silliest thing i've

heard note we can rewrite entropy just

like an expected value using fancy sigma

notation

the x represents a specific value for

surprise

times the probability of observing that

specific value for surprise

so for the first term getting heads

the specific value for surprise is 0.15

and the probability of observing that

surprise is 0.9

so we multiply those values together

then the sigma tells us to add that term

to the term for tails

either way we do the math we get 0.47

now personally once i saw that entropy

was just the average surprise that we

could expect

entropy went from something that i had

to memorize to something i could derive

because now we can plug the equation for

surprise in for x the specific value

and we can plug in the probability

and we end up with the equation for

entropy

bam

unfortunately even though this equation

is made from two relatively easy to

interpret terms

the surprise

times the probability of the surprise

this isn't the standard form of the

equation for entropy that you'll see out

in the wild

first we have to swap the order of the

two terms

then we use the properties of logs to

convert the fraction into subtraction

and the log of one is zero

then we multiply both terms and the

difference by the probability

then

lastly we pull the minus sign out of the

summation

and we end up with the equation for

entropy that claude shannon first

published in 1948

small bam

that said even though this is the

original version and the one you'll

usually see

i prefer this version since it is easily

derived from surprise

and it is easier to see what is going on

now

going back to the original example we

can calculate the entropy of the

chickens

so let's calculate the entropy for area

a

because six of the seven chickens are

orange we plug in six divided by seven

for the probability

then we add a term for the one blue

chicken

by plugging in 1 divided by 7 for the

probability

now we just do the math and get 0.59

note even though the surprise associated

with picking up an orange chicken

is much smaller than picking up a blue

chicken

there's a much higher probability that

we will pick up an orange chicken than

pick up a blue chicken

thus the total entropy 0.59

is much closer to the surprise

associated with orange chickens than

blue chickens

likewise we can calculate the entropy

for area b

only this time

the probability of randomly picking up

an orange chicken is 1 divided by 11

and the probability of picking up a blue

chicken is 10 divided by 11

and the entropy is

0.44

in this case the surprise for picking up

an orange chicken is relatively high

but the probability of it happening is

so low

that the total entropy is much closer to

the surprise associated with picking up

a blue chicken

we also see that the entropy value the

expected surprise is less for area b

than area a

this makes sense because area b has a

higher probability of picking a chicken

with a lower surprise

lastly the entropy for area c is one

and that makes the entropy for area c

the highest we have calculated so far

in this case even though the surprise

for orange and blue chickens is

relatively moderate one

we always get the same relatively

moderate surprise every time we pick up

a chicken

and it is never outweighed by a smaller

value for surprise like we saw earlier

for areas a and b

as a result we can use entropy to

quantify the similarity or difference in

the number of orange and blue chickens

in each area

entropy is highest when we have the same

number of both types of chickens

and as we increase the difference in the

number of orange and blue chickens we

lower the entropy

triple bam

p.s the next time you want to surprise

someone just whisper the log of the

inverse of the probability bam

now it's time for some

shameless self-promotion

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machine learning offline check out the

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