All language subtitles for 012 A Practical Example of Bayesian Inference_en

Instructor: Welcome to yet another practical example.

To make sure you understand

how all of these new terms are applied in practice,

we are going to go over some real-world data.

In particular, we will examine student population statistics

for Hamilton College during the 2017-18 academic year.

We will apply Bayes' Law to determine whether the college

is successfully diversifying its student population.

Since the college only provides a four year program,

any representation higher than 25%

would indicate a higher than average value.

Okay, to conduct our analysis

we are going to examine the Common Data Set

for Hamilton College for the 2017-18 academic year.

The Common Data Set, or CDS for short,

is a free public data set

of summarized statistics about the demographic

of the students attending a given college.

A new data set is released

after the start of every new academic year.

Furthermore, these sets are available for public use

and can be found either through the College Board website

or the site of the specific college you're interested in.

In this instance,

we visited Hamilton College's official webpage

and typed in CDS into the search bar.

Out of the results of our search

we chose the one for the 2017-18 academic year.

Great.

Now that you understand the data,

we can start off by opening

the CDS 2017-2018 PDF file attached to this lecture.

As you can see, section A is titled General Information

and includes basic information about the college.

Like contact details

and whether it is co-educational or not.

This information is important in general

but it is not relevant to our goal,

so we skip it and jump to section B on the third page.

This section is titled Enrollment and Persistence

and it showcases enrollment in the college

divided into different categories.

Now, that's something we'll make use of.

Let's start off by examining Table B1

which showcases the distribution of students

based on their gender.

Since the survey students fill out to apply

only has the options male and female,

the data is split solely into these two groups.

Now, let's spend some time ensuring

you interpret the table correctly

because we will be examining similar ones

throughout the lecture.

Start off by examining

the four different columns in the table.

They represent full-time and part-time students

separated by gender.

More precisely,

the first column showcases full-time male students

and the second one showcases full-time female students.

The third and fourth column

represent part-time male

and part-time female students, accordingly.

Now let us look at the rows.

We see that they're split into two major groups as well,

Undergraduates and Graduates.

Since Hamilton is a liberal arts college,

it is is an institution

which only offers bachelor's degrees.

There are not graduate students.

This explains why no numeric values

feature in the lower half of the table.

Furthermore, all claims we make

will be about undergraduate students.

Therefore, the number 217 in the table

is part of the first column and the first row.

Therefore, the people included

are simultaneously full-time males

and degree seeking first time freshmen.

In a Bayesian sense,

these 217 students represent the intersection

of the sets denoted in the first row and the first column.

Let's observe the total undergraduates row.

It includes all degree seeking

and all other students enrolled in credit courses.

This means that the value 1,000 we observe

in the sixth row of the second column,

is the union of all full-time undergraduate women

in the college.

Since there are no graduate students,

1,000 is actually the union

of all the full-time women in the entire college.

Now, the union of all women,

both full-time and part-time,

would be 1,000 plus nine or 1,009.

This is true

because no students can be simultaneously

part-time and full-time.

A Bayesian way of expressing this relationship

would be to say that the sets of part-time

and full-time students are mutually exclusive.

Additionally, we have 886 plus two, or 888 male students,

attending the college.

Below the table,

we are provided with the total number of enrolled students,

which is 1,897.

Since there are 1,009 women and 888 men,

they combine to complete the sample space.

Furthermore, due to the nature of the survey,

nobody was allowed to mark any answer different

from male or female,

so the two sets have no overlapping elements.

Satisfying both conditions

means the two sets are compliments.

All right, now that you're well acquainted

with how to read these tables,

we will move on to a different part of section B.

Namely, we will focus on the Racial/Ethnic diversity

in the college summarized in table B2 below.

We wanna use Bayes' Law

to determine whether the student body

is successfully diversifying its population.

Therefore, we need to use the table

to determine whether the freshman class

is more diverse than the average for the specific ethnicity.

To do so, we need to be able to accurately compute

the appropriate size of each set. we are interested in.

Let event A

be for a Degree-Seeking First-Time First-Year student,

and event B being Black or African American, non-Hispanic.

For convenience, we are going to refer to elements of A

simply as First-Years and elements of B as Black.

Since we have a total of 1,897 students,

and 480 of them are First-Years,

then the probability of being a freshman

equals 480 over 1,897 or 0.253.

This suggests that approximately 25.3%

of the student body are freshmen.

Similarly, we can estimate the probability

of a random student

at the institution being of African American descent.

That would equal 80 over 1,897 or 0.042, or close to 4.2%.

Now, the intersection of A and B

would represent all Black first year students.

Going back to the table,

only 26 students represent both demographics.

The probability of being a Black, First-Year student

is 26 over 1,897 or 0.014, which is close to 1.4%.

We know the likelihood of a student being African American

and we know the chance

of a random student being both Black and a freshman,

thus, we can use the conditional probability

to see that the likelihood of a Black student

being in his first year at the college

is 26 over 80, or 0.325.

This value is significantly greater

than the expected average of 0.25,

so we can see a rising trend

in the representation of minority in the student population.

The union of A and B represents all students

who are either First-Time, First-Years or Black.

We know that there are 481 First-Years.

80 Black and 26 First-Year Black students.

To find the number of students within the union of A and B,

we would apply the Additive Law.

According to the Additive Rule,

we would have 480 plus 80 minus 26, or 534 students,

that are either freshmen or Black.

Once again, we would find the probability

of being part of the union by dividing the size

of the union by the size of the sample space.

In this instance, that would be 534 over 1,897, or 0.281,

which indicates that approximately 28.1% of the student body

is either a freshman or identifies as Black.

So far so good.

Now, suppose C represents the set

of all Hispanic/Latino students at the college.

Since event B clearly says non-Hispanic,

then the two must be mutually exclusive.

Thus, the intersection of the two is the empty set

but their union equals the sum of their elements.

Therefore, according to the table,

there must be 167 plus 80, or 247 students,

who identify as either African American or Latino.

The probability of picking a random student

and them identifying as either one,

equals 247 over 1,897, or 0.13, which equals 13%.

Not that great as a percentage,

but great work on figuring that out.

You could surely find out these

in other relationships on your own.

However, let's dig a bit deeper

and examine some conditional probabilities.

In table B2, the entire first column

only represents values for First-Year students.

Therefore, any number we get

would represent the size of the intersection

of freshmen and another demographic.

This is important when we wish to compute

conditional probabilities.

Recall that the conditional probability formula

states that the likelihood of an event occurring

given another event has already occurred,

equals the likelihood of the intersection

over the likelihood of the second event.

A more precise example would be the following.

The likelihood of being Black,

given you are a freshman,

equals the probability of being a Black freshman

over the likelihood of being a freshman.

We can simplify this to the size of the intersection

over the size of the second set.

In our example, that would mean 26 over 480, or 0.054.

Therefore, there is a roughly 5.4% chance

for any freshman student to identify as Black.

Similarly, we can compute the likelihood

of a given student to be Hispanic, First-Year.

We can compute the likelihood of being a Latino,

given you are in your first year of college

as well as the likelihood of being a freshman

and apply the multiplication rule.

Let's begin.

We start by examining events A and C

being a First-Year and being Latino.

The likelihood of being Latino,

given you are a First-Year,

equals the number of Latino students who are First-Years

over all First-Years.

According to the table,

that equals 57 over 480, or 0.119, which is close to 12%.

So far so good.

Now that we've computed both probabilities,

we can plug the values into the multiplication rule.

The probability of being a freshman is 0.253

and the probability of being Latino,

assuming you are a First-Year, is 0.119.

By multiplying the two, we get 0.03,

or a 3% likelihood of being a Latino First-Year.

Great job.

What if we wanna find out the likelihood

of being a freshman, given you are Hispanic?

We could calculate this using two different ways.

In the first one,

we would simply apply the conditional probability formula

like we did earlier.

However, we could also apply Bayes' Law to solve this.

According to the theorem,

the likelihood of being a freshman,

assuming you are Hispanic,

equals the likelihood of being Latino,

given you are a First-Year,

times the probability of being a freshman

over the probability of being Hispanic.

Next, we estimate the likelihood of being Latino,

which is 167 over 1,897, or 0.089, and that is close to 9%.

We have estimated all three of the required probabilities

and they are respectively equal to 0.119,0 .253, and 0.089.

Plugging these values into the formula

gives us 0.199 times 0.253 divided by 0.089, or 0.338.

That means there is a 33% chance

a student is a First-Year assuming they are Hispanic.

Thus, we can say that a person is more likely

to be a First-Year, given they are Hispanic,

than to be Hispanic, given they are a freshman.

If we think about the favored overall formula,

this makes sense because there are more freshmen

than Hispanic students in the college.

Such a characteristic is fairly common

among small liberal arts colleges in upstate New York,

so the insight does not surprise us.

Phenomenal work.

We examined several tables from the Common Data Set

for Hamilton College for the 2017-18 academic year

and our short analysis suggests

that the college is improving its minority representation

with the current freshman class.

However, further research would be required

to account for attrition among the student population

as well as moving to other colleges within the region.

Even though our analysis

may not have been full or conclusive,

we made full use of our understanding of Bayesian notation

to reach some insight about the data.

This shows how important Bayesian inference is

in terms of analytics

and how understanding the relationship

between sets and events

can help us reach important conclusions.

For homework, you can explore section C of the CDS

which summarizes First-Time, First-Year admissions.

Compute what the likelihood of a First-Time male student

to be accepted, based on gender,

and determine whether being male or female

had any effect on your chances of acceptance.

Furthermore, determine whether First-Time freshman women

were more likely to enroll

than first time freshman men.

To find both of these,

you need to examine only the tables in part C1 of the CDS.

Additionally, you can practice your understanding

of probabilities by exploring the values

in table C2 and determining the likelihood

of being offered a place on the wait list.

Additionally, you can compute the chance of being admitted

having accepted a place on the wait list

and the likelihood of getting admitted

given you are offered a place on the wait list.

In the next section,

we are going to start talking about

probability distributions,

how to properly apply them

and why understanding the most commonly featured ones

is so important.

Thanks for watching.