Naive Bayes classifier: A friendly approach

Summary

In this video, Luis Serrano explains the Naive Bayes classifier, a fundamental concept in probability and machine learning. He uses the example of building a spam detector to illustrate how Bayes' theorem is applied. The video covers calculating the probability of spam emails based on keywords like 'buy' and 'cheap'. It also discusses the 'naive' assumption of independence between features, which simplifies calculations. Luis provides a detailed walkthrough of how to apply Bayes' theorem and the naive assumption to estimate probabilities when not all data points are available.

Takeaways

• 📝 Bayes' Theorem is a fundamental concept in probability and machine learning, used to calculate the probability of an event based on prior knowledge of conditions.
• 📝 Naive Bayes is an extension of Bayes' Theorem that simplifies calculations by making the assumption that features are independent, even when they might not be.
• 📝 The video uses the example of a spam detector to explain how Naive Bayes can be applied to classify emails into spam or not spam based on the presence of certain words.
• 📝 The script demonstrates how to calculate the probability of an email being spam if it contains specific words, like 'buy' and 'cheap', using Bayes' Theorem.
• 📝 It explains the concept of conditional probability and how it is used in the context of Naive Bayes to determine the likelihood of spam based on email content.
• 📝 The video highlights the importance of making naive assumptions about independence between features to simplify the calculations and make the model more manageable.
• 📝 The script shows how to handle situations where data is sparse or certain combinations of features do not appear in the training set.
• 📝 It emphasizes that even with the naive assumption of independence, Naive Bayes classifiers can perform well in practice for many classification tasks.
• 📝 The video concludes by summarizing the process of filling out a probability table and using it to calculate the likelihood of an email being spam based on multiple features.
• 📝 It challenges viewers to understand the math behind Naive Bayes and appreciates the simplicity of calculating probabilities by dividing one set of data by another.

Q & A

What is the Naive Bayes classifier?

The Naive Bayes classifier is a simple probabilistic classifier based on applying Bayes' theorem with strong (naive) independence assumptions between the features.

What is Bayes' theorem?

Bayes' theorem is a fundamental principle in probability theory that describes the probability of an event based on prior knowledge of conditions that might be related to the event.

How does the Naive Bayes classifier work for spam detection?

For spam detection, the Naive Bayes classifier works by calculating the probability of an email being spam based on the presence of certain keywords or features that are indicative of spam.

What is the significance of the word 'buy' in the context of the spam detector example?

In the spam detector example, the word 'buy' is chosen as a feature that is likely to appear more frequently in spam emails compared to non-spam emails.

How is the probability of an email being spam calculated if it contains the word 'buy'?

The probability is calculated by dividing the number of spam emails containing the word 'buy' by the total number of emails containing the word 'buy'.

What is the role of the word 'cheap' in the spam detection example?

Similar to 'buy', 'cheap' is another feature that might be more common in spam emails, and its presence is used to calculate the likelihood of an email being spam.

What happens when you apply Naive Bayes to multiple features, like both 'buy' and 'cheap'?

When applying Naive Bayes to multiple features, you calculate the combined probability of an email being spam given the presence of all those features, assuming independence between them.

Why is the assumption of independence between features considered 'naive'?

The assumption of independence is considered 'naive' because in reality, features are often not independent. However, this simplification allows for easier calculations and can still yield good results.

How does the Naive Bayes classifier handle situations where certain combinations of features have not been observed in the training data?

The classifier uses the assumption of feature independence to estimate probabilities for unseen combinations, allowing it to make predictions even with limited data.

What is the importance of the dataset size when using the Naive Bayes classifier?

A larger dataset can provide more accurate probabilities for the features, but the Naive Bayes classifier can still perform well with smaller datasets due to its simplicity and the assumption of feature independence.

Can the Naive Bayes classifier be improved by considering feature dependencies?

Yes, the classifier can potentially be improved by using more sophisticated models that capture feature dependencies, but this comes at the cost of increased complexity and computational requirements.

Outlines

00:00

📊 Introduction to Naive Bayes Classifier

Luis Serrano introduces the concept of the Naive Bayes classifier, explaining its importance in probability and machine learning. He clarifies that Bayes' theorem is about calculating the probability of an event given some prior knowledge, and Naive Bayes extends this idea by making simplifying assumptions to handle complex scenarios. The example of building a spam detector is used to illustrate the concept, where the presence of certain words in emails is correlated with them being spam or not. The video uses the word 'buy' to demonstrate how to calculate the probability of an email being spam based on its content.

05:00

🔍 Handling Overlapping Features in Naive Bayes

The script delves into the challenge of handling multiple features, such as the words 'buy' and 'cheap', in a Naive Bayes classifier. It discusses the issue of zero instances of non-spam emails containing both words in a small dataset and how this could skew the classifier's accuracy. The solution proposed is to make an assumption about the independence of these features and estimate the probability of their co-occurrence based on their individual probabilities, despite the lack of direct evidence in the data.

10:00

📉 Applying Naive Assumptions to Improve Calculations

Luis explains how the Naive Bayes classifier uses the assumption of independence between features to simplify calculations. By assuming that the presence of one word does not affect the presence of another, the video demonstrates how to estimate the probability of an email being spam based on multiple keywords. It shows how to calculate this probability by multiplying the probabilities of individual words appearing in spam emails and compares it to the same calculation for non-spam emails.

15:01

📈 Expanding Naive Bayes to More Features

The script extends the discussion to include more features, such as the word 'work', in the Naive Bayes classifier. It shows how to incorporate additional features into the model by making the same naive independence assumption and calculating the combined probability of multiple words appearing in an email. The video explains how some features can increase the likelihood of an email being spam, while others can decrease it, and how the Naive Bayes classifier combines these to make a prediction.

20:02

📝 Wrapping Up Naive Bayes Explanation

In the final paragraph, Luis summarizes the process of using Naive Bayes for spam detection, emphasizing the simplicity of calculating probabilities by dividing one number by another. He invites viewers to engage with the content by subscribing, liking, sharing, and commenting with questions or suggestions for future videos. The video concludes with a reminder to follow Luis on Twitter for more mathematical insights.

Keywords

💡Naive Bayes Classifier

The Naive Bayes Classifier is a simple yet powerful probabilistic classifier based on applying Bayes' theorem with strong (naive) independence assumptions between the features. In the video, Luis Serrano uses the example of a spam detector to illustrate how the classifier works. The classifier calculates the probability of an email being spam based on the presence of certain words, like 'buy' and 'cheap', assuming that the presence of these words does not depend on each other.

💡Bayes' Theorem

Bayes' Theorem is a fundamental principle in probability theory and statistics that describes the probability of an event based on prior knowledge of conditions that might be related to the event. In the context of the video, Bayes' Theorem is used to calculate the likelihood of an email being spam given the presence of certain words. The theorem is fundamental to understanding how the Naive Bayes Classifier makes its predictions.

💡Spam Detector

A spam detector is a system designed to identify and filter out unsolicited emails, often referred to as 'spam'. In the video, Luis Serrano uses the spam detector as an example application of the Naive Bayes Classifier. The goal is to sort emails into 'spam' or 'ham' (non-spam) categories based on the presence of certain keywords.

💡Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has occurred. In the video, conditional probability is used to calculate the likelihood that an email is spam based on the presence of specific words like 'buy'. For example, the probability that an email is spam given that it contains the word 'buy' is calculated as 80% based on the dataset provided.

💡Feature

In machine learning, a feature is an individual measurable property or characteristic of a phenomenon being observed. In the video, features refer to properties of emails such as the presence of specific words ('buy', 'cheap', 'work') that are used to predict whether an email is spam or not.

💡Independence Assumption

The independence assumption in the context of the Naive Bayes Classifier is the assumption that the presence of a word is unrelated to the presence of any other word. This simplifies the calculations significantly but may not always reflect reality. In the video, Luis Serrano explains that assuming 'buy' and 'cheap' are independent makes the math easier, even though they might not be in reality.

💡Dataset

A dataset is a collection of data, typically used for analysis or to train machine learning models. In the video, Luis Serrano refers to a dataset of 100 emails, with 25 marked as spam and 75 as non-spam, which is used to train the spam detector and to illustrate how the Naive Bayes Classifier works.

💡Probability

Probability is a measure of the likelihood that an event will occur. In the video, probability is used extensively to determine the likelihood of an email being spam based on the presence of certain words. The calculations involve ratios and percentages derived from the dataset to estimate these probabilities.

💡Email

An email, short for 'electronic mail', is a method of exchanging messages and information from an author to one or more recipients. In the video, emails are the subject of classification, with the Naive Bayes Classifier being used to distinguish between spam and non-spam (ham) emails.

💡Ham

In the context of email filtering, 'ham' refers to legitimate, non-spam emails. The term is used in contrast to 'spam', which refers to unsolicited or junk emails. In the video, Luis Serrano uses the terms 'spam' and 'ham' to categorize emails in the dataset used to train the Naive Bayes Classifier.

Highlights

Naive Bayes classifier is based on Bayes' theorem and is useful in machine learning for tasks like spam detection.

Bayes' theorem is about calculating the probability of an event given some information about another event.

Naive Bayes simplifies calculations by making assumptions about the independence of events.

An example of building a spam detector using email data is provided.

The word 'buy' is studied for its correlation with spam emails.

It's found that 80% of emails containing 'buy' are spam.

The word 'cheap' is also studied, with 60% of containing emails classified as spam.

When considering both 'buy' and 'cheap', the probability of an email being spam reaches 100%.

The naive assumption is made that words 'buy' and 'cheap' are independent.

The independence assumption allows for easier calculations even with limited data.

The concept of 'ham' is introduced as a term for non-spam emails.

The video explains how to fill out a probability table using Bayes' theorem.

The importance of normalization in calculating final probabilities is discussed.

Naive Bayes can handle many features by assuming independence between them.

The video concludes by emphasizing that Naive Bayes combines multiple features into a model for spam detection.

The presenter invites viewers to engage by subscribing, liking, sharing, and commenting for more content.

Transcripts

i am luis serrano and this video is

about the naive Bayes classifier now

your base is one of the most important

things in probability and it's very

useful in machine learning you may have

seen it as a complicated formula

regarding some ratios of probabilities I

like to see this a little further and I

like to think of it as what is the

probability of something happened given

that we know some information that

something else happens and then naive

Bayes is an extension of this which

basically says ok once I have too many

events and I don't know how to handle

them are there any naive assumptions

that I can make on them to make the math

work easier and so this is what we're

gonna see today so let's start with an

example let's say we want to build a

spam detector because we are tired of

seeing a lot of spam email in our inbox

and we want to sort it properly so how

do we build it we build it with previous

data unless our previous data is a set

of a hundred emails and when we look at

them carefully there are 25 of them that

are spam and 75 and are not spam

so what we're gonna do is we're gonna

try to pick properties of the emails

that we think may correlate with them

being spam or not spam so let's pick one

let's say we're gonna study the

appearance of the word buy so we think

that emails that contain the word buy

are more likely to be spam than not spam

so let's study that let's see how many

emails that a spam have the word Buy and

turns out there's 20 of them and let's

see how many emails that are not spam

have the word buy on them

so there's five so let's forget about

all the others and just look at the spam

emails and here's a quiz the quiz says

if an email contains the word buy then

what is the probability that this email

is spam given the data that we have and

the options are 40% 60% 80% and a

hundred percent so feel free to pause

the video and think about it yourself

given the data that we have what is the

probability that if an email contains

over by then it is spam is a conditional

probability so I'll tell you the answer

the answer is if we look at the emails

that contain the word buy well there's

$20 spam and five that are not

so that mason 80/20 split and so from

this data we can see that from the

emails our continued whereby 80% of them

are spam so the probability we conclude

again just from this data that the

probability is gonna be 80 percent that

it's spam if it contains the word buy

therefore we associate the condition

containing the word buy with the

probability 80 percent and that is

exactly what Bayes theorem is you may

have seen in a different way it's you

know like a formula this is really what

it is so just for fun let's do it for a

different property for a different word

let's say that we think that the word

cheap may also be a good way to tell if

an email is spam so let's study this

word we count how many times the word

cheap appears in spaniels that's gonna

be in 15 of them and from the non-spam

ten of them I have the word cheap so we

forget about the rest and quiz again if

an email contains where chip was a

probability a spam 40 60 80 100 again

feel free to pause the video I'll tell

you the answer the answer is 60% because

if you look at the split there is 15

spam and 10 no spam among the ones that

contain the word cheap so that's a 60/40

split and therefore the solution is 60%

so we applied base theorem for two words

and obtain 80 and 60 now here's where

things get complicated what if we want

to apply it for both words at the same

time so we want to see what's the

probability of an email being spam if it

contains both the word bye

and the word cheap well we can do the

same thing right we can count how many

emails contain the word by and then look

at how many contain the word cheap and

then actually look at the overlap and so

there's actually 12 emails that contain

the words buy and cheap so that's some

good data and then let's look among the

no spams let's say that there's these

five that contain the word buy and these

ten contain they were cheap so actually

there's none that contain both words but

that's okay we're gonna do the same

thing as before we have 12 spam emails

and zero no spam emails that contain

they were cheap so easiest quiz in the

world if an email contains the word

buying cheap wise

probably a spam forty sixty eighty or a

hundred and this should be easy right

because there are twelve emails that

contain both words zero emails that

contain no words and this is a 100% 0%

split so the answer is 100% and we are

done

right well maybe you're being skeptical

like me right it seems like that's a

little too much like any classifier that

tells you something 100% is too strong

and where lies the problem well the

problem lies here that we had 12 emails

that contained about words by and cheap

and that's not bad but here we had 0

emails so among the non spam emos there

are zero emails that contains the words

buy and cheap and so that's just

unfortunate among our data we don't find

the two words but it's possible that

these two words could appear right so we

can't restrict ourselves to not have a

classifier with the words buying cheap

just because in our small data set the

world stone appear so what could we do

well one solution could be just maybe

collect more data like go through a lot

more emails until we find the words buy

and cheap and then do base theorem on

those but what if we just can't what if

we can't collect more data and we have

to do with the data that we have so

let's think we have this situation what

would you do if you have the situation

and you have to sort of imagine how many

emails would contain the words buy and

cheap so what we're gonna do is try to

guess the number try to come up with a

sensible amount of emails that would

contain the words buy and cheap even if

we found none so let's look at a

slightly larger DSL let's say we have a

hundred emails so this is a different

set than the first one we have a hundred

emails and let's say that five contain

the word buy and let's say that ten

contain the word cheap and they don't

overlap however what do you think would

be a sensible number of emails that

would contain the words buy and cheap so

let's think 5 out of 100 is 5% so 5% of

the emails contain the word body and 10

out of 100 is 10 percent so tempers

the emails contained that were cheap so

in an ideal world where everything was

pretty how many emails would contain the

words buy and cheap well what is what is

ten percent of five percent it's zero

point five percent so why don't we just

assume that 0.5% of the email contained

the word buy and cheap so we can sort of

imagine that there is half an email that

contains the words buying cheap answers

all we're doing is math it doesn't

really matter that there's half an email

this will work out on our formulas what

we did is an assumption we assumed that

the words buy and cheap are independent

they may not be right it could be that

containing the word buy makes it easier

to contain the word cheap because you're

talking about a product they say buy

cheap something or it could be the

opposite that if one appears that sort

of forces the other want to do not

appear to be less likely to appear so

it's a it's a quite a strong assumption

as a matter of fact many people would

say that's a naive assumption assuming

that two variables are independent when

they may not be is very naive however

that's what our algorithm is based

because it turns out that if we make

these assumptions things still work well

and it makes our math much much easier

because now we don't have to collect

thousands of emails we can collect these

100 and from the number of thousands by

and the number of peers of cheap we can

sort of cook up the numbers of Pearson's

of buy and cheap so let's do that let's

go back to our data we had 25 spam

emails and 20 of them had the word buy

and that's four-fifths and 15 of them

had the word cheap that's 3/5 so we can

imagine that the product of this is 12

divided by 25 so we could assume that an

average 12 emails here out of 25 would

contain the words buy and cheap so in

order to find the actual number we

multiply by 25 and we get that 12 emails

have the words buying cheap so that was

kind of lucky that we actually did find

12 we're not gonna be that lucky in the

other case but we can still do it right

so we have 75 emails five of them are

buy that's 115

of them then ten of them have the word

cheap that's two fifteen of them and the

product of these two fractions again

assuming they're independent is two

divided by 225 so that's the fraction of

emails that contain the words buy and

cheap so to find the actual number or

multiply it by 75 and we get 2/3 so in

here we have 2/3 of an email contains

the words buy and cheap and that's fair

let's work with that so we go back to

our data and on the Left we have 12

emails that contain the word buy and

cheap and on the right we have 2/3 of an

email that contain the word buy and

cheap and we can do math with these ones

right because now the quiz says if an

email contains the words buy and cheap

what is the probability that is spam so

let's do some math what is the split

among 12 and 2/3 well let's take the

spam ones that's 12 and let's take the

total number of emails that contain Buy

and cheap and that's 12 plus 2/3 because

there's 12 spam and 2/3 there are no

spam so we can find the ratio between

these and by the way if you've seen the

formula for base theorem and there's a

ratio and it's precisely this one so

what do we do with this fraction well we

put in lowest terms is 36 over 38 or

ninety four point seven three seven

percent because this plate is ninety

four point seven three seven and five

point two six three therefore our final

answer is that the words buy and cheap

give us a probability of ninety four

point seven three seven percent of being

spam that means if we have an email with

both of those words is ninety four

percent point seven three seven likely

to be spam and that is precisely the

naive Bayes classifier so now Bayes

classifier basically it's a combination

of Bayes theorem and be naive assumption

that two events are going to be

independent when they may not be but

that naive assumption makes the math

much much easier so let's do a little

summary what we're really doing is we're

gonna fill out this table and some

places of the table we can't really fill

out the data so we'll fill them out with

other places in the table so let's look

at spam and those

animals we looked at the total was 25

spam emails and 75 non-spam emails in

our dataset right now the next way we're

gonna count how many of them have the

word by so 20 of the 25 have the word by

that's forfeits and then five of the 75

there are no spam have the word by so

that's 115 because it's five divided by

75 now we're gonna fill in the next row

so out of the spam emails the 15 of them

contain the word sheep that's

three-fifths cousins fifteen by twenty

five and ten under $75 not spam contain

they were cheap and so that's two 15s

because it's 10 divided by 75 now we

would love to fill in the last row with

data the word their words buy and cheap

but unfortunately this is not big enough

to actually handle as an event that is

so sparse like the words buying cheap

appearing and you can imagine if there

were more words it would be even harder

so we have to cook up this row from the

previous ones so what we're gonna do is

the naive assumption that the words buy

and cheap are independent so that one

doesn't imply or push the other one to

appear or stop it from appearing and if

we make this assumption then we're gonna

say that the product of these two is the

probability of the word buy and cheap

appearing so that's 12 divided by 25 the

product of 4/5 and 3/5 so that's gonna

be our probability and now if this is

the probability of buying cheap

appearing how many emails contain buy

and cheap all we have to multiply by it

by the total number which is 25 so 25

times 12 over I 25 is 12 so we conclude

that 12 we must should contain the words

buying cheap even if there is 12 or 14

or 10 or none logically if we have that

assumption there should be 12 now let's

look at the other two boxes well again

we make the assumption that the word

pine and chips are independent of each

other so the product of this 2 which is

2 divided by 225 it's gonna be the

probability afterwards buying cheap

appearing in an email that is no spam so

now how many emails that are not spam

contain the word buy and cheap well

product of the probability times a total

number so how much is two over twenty

two hundred twenty five times seventy

five that's actually two-thirds so we

have twelve spam emails and two-thirds

of an email that is not spam that

contain the words buying cheap so now we

have to normalize right we have to see

what is the split how many percentage

are spam among the total ones and the

total ones is twelve plus two-thirds

that's all of our emails that are

containing the word pion cheap so we

divide twelve the spam ones divided by

the total which is twelve plus

two-thirds and we get 36 over 38 which

is nine four point seven three seven now

notice that nice ways extents and the

idea is that this extends to many many

more properties right because the point

is if we have 50 properties and we can't

check when they all appear at the same

time we can check when one appears and

then multiply things right so let's add

an extra row to this table let's say we

looked at the word work and we're

wondering if the word work helps us in

our classifier so let's study how much

it appears let's say that it appears

five times in our spam emails and 30

times in our non-spam email so it

doesn't look like it's gonna help us

that much it looks almost like it's a

word that's more correlated to not spam

but let's just study it so this 5 out of

25 is 1/5 so therefore one fifth of the

spaniels contain the word work and six

fifteen of the nonce problems contain

the word work because 30 divided by 75

is 6 over 15

so again naive assumption that the words

buy cheap and work are all independent

therefore the probability that the three

of them appear in an email is the

product of these three numbers which is

12 divided by 125 and again if we want

to estimate the number of emails that

are spam that contain those three words

we multiply the probability times the

total and we get twelve divided by five

so roughly twelve divided by five which

is a little over two emails will be spam

and contain the words by chip and work

and now let's do it over here we assume

again that the three words are

independent of each other we take the

product of the probabilities and that's

gonna be the probability that the words

buy cheap and work all appear in an

email at the same time when the email is

not spam

so in order to find the total number of

emails that are not spam to contain the

words by cheap and work we multiply the

probability that they appear times the

total number of emails and we get that

four out of 15 emails are not spam and

contain the word by Cheban work because

75 times 12 divided by three three seven

five is four fifteen so in summary out

of the emails that contain the words by

chip and work 12 over five are spam and

four over 15 or ham so how many are spam

divided by the total well we take twelve

hour by five the number spam divided by

the total which is 12 over 5 plus four

divided by 15 and that is gonna be 36

over 44 put in lowest terms or 90% so

that's how we combine the three words

now is that 90 is less than 97 because

the word work actually decreases the

probability that an email is spam

because as you can see work appears a

lot more in spam emails so it does make

sense because it's not a word that one

would correlate with spam so some of

these properties may increase

probability and some of them would

decrease it but the fact is a nice base

helps us combine a bunch of different

features into creating a model that

calculates the probability that

something is spam and these features get

combined in a nice way because we don't

have to wait until we find an email with

all these features we can actually cook

up probabilities without having emails

that satisfy all of them so if you're

like formulas this is really what

happened in the background we have this

is the formula of Bayes theorem and the

letter S stands for being spam the

letter H stands for ham which is

actually how they call email that are

not spam they call them ham and the red

letter B stands for by so probability of

s given B when you see that vertical bar

that is a conditional probability so

what the Left says is probability of

spam in the word by appears and that's a

ratio because most post probabilities

are ratios and then the top we have

probability of BI given that spam so out

of the spanning knows how many of them

contain the word by that was 20 out of

25 and then probability of s is

email spam regardless of any words that

it contains to us 25100 because if we

remember there were 25 spaniels out of

100 total so in the bottom goes

everything that total so that's the same

thing 20 over 25 times 25 or 100 plus

the ham ones so we have what's the

probability of the word by appearing if

the email is ham that's five or seventy

five because out of 75 animals five of

them have the word pie and the

probability of animal being ham well 75

over 100 so if you do that whole formula

you get 80% but the interesting thing is

if you look at what we did it was

exactly that and then what happens with

naive Bayes is that we make this

assumption that the probability of the

word by and the word cheap appearing is

the product of the probabilities of the

word by appearing and the word cheap

appearing again this is not supposed to

happen the words buying cheap may be

either correlated or inversely

correlated maybe one implies the other

one maybe one stops the other from

appearing but we're gonna assume naively

that the product of the probabilities

the property of both appearing which is

saying this that the probability of some

event B intersection segment C is a

product of probabilities of B and C

appearing again is a naive assumption

but we're gonna make it because I've

makes our math easier and the full

formula for a naive Bayes this is for

two events but you can generalize this

for many more events is probability of

spam given that the words buy and cheap

appear is that formula and if we look at

all the probabilities here we say

probability of spam if it contains the

words buy and cheap well it's a ratio on

the top we know these probabilities is

20 out of 25 or probability of by given

that a spam probability of cheap given

that it's spam is 15 or 25 you remember

correctly or 15 e spam emails containing

the word cheap and then again 25 over

100 for the probability that an email of

spam in the bottom we have the same

thing plus 5 over 7 5 the probability

that a ham email contains the word pie

10 over 75 the product is at honey milk

contains were cheap and then the

probability that an email is ham which

is 75 over 100 you do this math and you

get ninety four point seven three seven

but I challenge you if it doesn't look

super clear look at this slide and go

to what we did in night base and

convince yourself this is exactly what

we did

what do then this whole video was

nothing different than calculating

probabilities by dividing one thing by

another so thank you very much that's it

for a naive base as usual if you liked

it please subscribe for more videos

coming up yeah please hit like share it

with your friends and feel free to

comment to ask any questions or any

suggestions for this or any other videos

you'd like to see and my twitter handle

is louis likes math so thank you very

much for your attention and see you in

the next video