Entropy (for data science) Clearly Explained!!!
Summary
TLDRThis 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
📊 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.
🧐 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.
📉 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.
🔄 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.
Mindmap
Keywords
💡Entropy
💡Data Science
💡Expected Values
💡Classification Trees
💡Mutual Information
💡Relative Entropy
💡Cross Entropy
💡Surprise
💡Logarithm
💡Coin Flip
💡Dimension Reduction
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
if you want to review statistics and
machine learning offline check out the
statquest study guides at statquest.org
there's something for everyone
hooray we've made it to the end of
another exciting stat quest if you like
this stat quest and want to see more
please subscribe and if you want to
support statquest consider contributing
to my patreon campaign becoming a
channel member buying one or two of my
original songs or a t-shirt or a hoodie
or just donate the links are in the
description below
alright until next time quest on
5.0 / 5 (0 votes)