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Hello and welcome back to the course
on Machine Learning.

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In today's tutorial,
I'm going to show you my solution

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of the challenge
that I threw at you Lausanne.

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So in the previous tutorial,
we had the challenge

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to calculate the posterior
probability of somebody being a person

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that drives to work a
given that they are placed in the position

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where we are placing the new observation
for our data set.

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So basically, what is the likelihood

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of that new observation
representing a person who drives to work?

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And the way we need to
calculate is using the Bayes theorem,

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which is right in front of us.

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As I know we're going to walk through it.

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First we're going to calculate the prior
probability.

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Then we're going to calculate
the marginal likelihood.

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And then we're going
to calculate the likelihood.

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And you can compare along the way
if you got the same result.

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So let's go through it.

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So there's our data set.

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And now let's move it to the left
to make some space.

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And the first thing we're going to
calculate is a prior probability.

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The prior probability.

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There's kind of two ways
to think about it.

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So the first way to think about it
is if I just randomly select

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a person from our data set right now.

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So not including the gray dot,

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not including the new data point,
if I randomly select a person from there,

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what is the likelihood of them
being a person who drives to work right?

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So that will be just
the number of the answer to

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that is the number of green dots
or the total number of dots.

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The other way to think about it
is if I just randomly throw

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a new data point into our data set, right?

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Just randomly and without knowing anything
about their age or salary,

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then just knowing all of this prior
information that we have the green dots

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and the red dots, what is the likelihood
of that person that we're adding?

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What is their likelihood
to be a person who drives to work?

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Again, we don't have is very simple
because we don't have any other choice

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but to calculate the probability
and assign the probability

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just based on what we know.

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And that is just to take the green dots,

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add the 20 green dots
divided by the total number of dots.

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Here, and assign that as a probability.

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So we don't have any other choice.

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And therefore that's
what it's calculated as.

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So the probability of somebody
being a person who drives to work

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or a random dot
selected out of our existing dots,

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being a person who drives to work
is a number of drivers,

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which is 20 divided by total duration,
which is 30.

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So we go 20 over three.

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So that was the prior probability
we've done that.

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Next one is a marginal likelihood.

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So let's go ahead and calculate that.

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And you'll find that
the marginal likelihood

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is actually going to be exactly the same
as in the previous tutorial.

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And we'll talk about this separately.

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So again we're going to draw the circle
around our observation.

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We're to remove observations.
So it's not in the way.

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Then we're going to shade in this area.

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And so now the marginal likelihood
is the question what is the likelihood of

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if I just pick a random dot from our data
set just randomly.

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What is the likelihood
that I'm going to pick one out of here.

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So what the reason why we put X here is,
is because what is the likelihood of me

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picking a observation
that exhibits features

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similar to the features of that point
that we are adding to our dataset?

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So the point we're adding to the data set
we've just removed, it is over there.

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And we've agreed that any dot inside
the circle is deemed to be similar

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to that dot, or in other words,
deemed to be exhibiting similar features

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so similar age
and similar salary to that dot.

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And therefore that's
what we're calculating.

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So P of X is very simple.

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We just need to calculate the number
of similar observations,

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the number of observations
that actually fall in here, which is four

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divided by the total number
of observations which is 30.

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So that will give us the likelihood
of a new dot falling here,

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or the likelihood of if we just pick out
a random dot from our dataset right now,

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then the likelihood of it
being one of these is four over 30.

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There we go for all 30.

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All right.

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So that is our marginal likelihood done.

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And now we're going to move on
just the likelihood.

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And this time is going to be
the likelihood of somebody exhibiting

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the features of X or being similar
to the datapoint that we're adding,

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given that we're only looking at people
who are driving to work.

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So let's have a look at that.

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Here's our dataset.

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And again we're going to draw the circle
around our data point.

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Take it out and then add that shading.

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So now the question is given
that we're only dealing

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with people who drive to work,
what is the likelihood that if we pick

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one of them, that that person
will be exhibiting features similar to X?

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So because we're only dealing
with the people who drive to work,

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we can forget about the red dots.

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There you go.

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This shaded there faded out and now
we're only dealing with the green dots.

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So the question is,
given that we're selecting a random point

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out of all of the people that drive.

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So this vertical bar drives
means given that a person drives to work.

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So we're looking at a random point
out of these.

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What is the likelihood that they will
exhibit features similar to X,

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which we agreed is the same
as they fall inside this circle.

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And the likelihood of that is one over
the total number of green dots.

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So there we go
p of x, given that they drive to work,

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is the number of similar observations
among those who walk.

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So inside of here is one

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similar meaning similar to our new points
that we're adding.

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And then divided
by the total number of workers.

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And that's not 20.

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So one over 20 that is our likelihood.

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There we go.

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So now we can plug these into the formula.

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Calculate the posterior probability.

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So it's going to be one over 20 times
20 over 30 divided by four with 30.

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And it's 0.25.

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So 25%.

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So there we go.

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That was step two of our Naive

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Bayes algorithm or Naive Bayes classifier.

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Hopefully you were able to follow along.

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And also that I hope that you had chance

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to perform that exercise on your own
and you got a similar result.

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And that's it for today.

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I look forward to seeing you next time.

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And until then, enjoy machine learning.