Original subtitles

Instructor: Hello again.

In this lecture, we are going to introduce

one of the most important formulas

in the world of probability.

The Bayes' rule.

People also refer to it as Bayes' Theorem, or Bayes' Law.

So we will use all three interchangeably.

For starters, take two events, A and B.

According to the conditional probability formula,

the conditional probability

of A given B equals the probability

of their intersection over the probability of event B.

Using the multiplication rule,

we can transform the numerator

of this fraction to get the probability of the intersection

of A and B equals the conditional probability

of getting B given A times the probability of getting A.

Therefore, the conditional probability

of getting A given B is equal to the following fraction.

The conditional probability

of getting B given A times the probability of A

divided by the probability of B.

This equation is known as Bayes' Theorem.

It is crucial because it allows us to find a relationship

between the different conditional probabilities

of two events.

One of the most prominent examples of using Bayes' Rule

is in medical research when trying to find

a causal relationship between symptoms.

Knowing both conditional probabilities

between the two helps us

make more reasonable arguments

about which one causes the other.

For instance, there is certain correlation between patients

with back problems and patients wearing glasses.

More specifically, 67% of people with spinal problems

wear glasses while only 41% of patients with eyesight

issues have back pains.

These conditional probabilities suggest that it is much more

likely for someone with back problems to wear glasses

than the other way around.

Even though we cannot find a direct causal link

between the two, there exists some arguments

to support such claims.

For instance, most patients with back pain

are either elderly or work a desk job

where they remain stationary for long periods.

Old age and a lot of time in front of the desktop computer

can have a deteriorating effect on an individual's eyesight.

However, many healthy and young individuals wear glasses

from a young age.

In those cases, there is no other underlying factor

that would suggest incoming back pains.

All right, similarly,

we can also apply Bayes' Theorem in business.

Let's explore this fictional scenario.

Your boss wants you to do research about what companies

are looking for in recent college graduates.

Good academic performance or working experience.

You go over the resumes of the last 200 people

who matched the requirements

and got the job they applied for.

Out of those candidates, 45% had the relevant experience.

In addition, 60% had good grades.

Furthermore, we know that out of those 45%

who had relevant experience,

50% also performed well academically.

To avoid any harsh decisions,

we need to compute the conditional probability

of the candidate to have relevant experience

provided they had a high GPA.

If we used Bayes' Theorem,

we get that the conditional probability

of a candidate performing well academically

to have relevant experience is 0.5 times 0.45

over 0.6 or approximately 0.375.

Since 0.5 is greater than 0.375, then it is more likely

for an experienced candidate to excel academically

than for a student with high grades

to have the required working pedigree.

Thus, candidates who had internships are more likely

to also have a high GPA.

Therefore, firms are much more likely

to get their ideal candidate if they go for somebody

who has experience rather than somebody

who thrived academically.

Well, that explains a lot.

Good thing, online courses

are a completely different category.

To wrap this section up, let's briefly go over the odd case

of applied Bayes' Theorem to independent events.

Take these two events,

the weather being sunny and your code not working.

You can always try and blame it on the sun being too bright

but deep down, you know the weather has nothing to do

with your code not compiling.

Similarly, rain, wind, and snow are seldom affected

by how well your algorithm performs.

Let's get more specific.

Imagine, you know the following,

the probability of your code working is 0.3

and the likelihood of it being sunny tomorrow is 0.4.

Since the two events are independent,

the probability of sunshine

provided your code works is also 0.4.

If we plug these values into Bayes' Theorem,

then we get that the likelihood of your code working

in good weather equals 0.4 times 0.3

divided by 0.4 or simply 0.3.

This result aligns with our suspicion

that the chances of your algorithm performing as intended

neither increase, nor decrease based on the weather.

Good job, everybody.

In the next section of the course,

we are going to focus on probability distributions.

We will introduce the most important ones.

Talk about their expected values

and how wide their prediction intervals should be.

Thanks for watching.