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Hello and welcome back to the course.

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Today
we're talking about maximum likelihood.

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So we have this curve
that's fitting our data.

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It's great.

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But how do we know that this is the best
curve that can fit our data.

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Just like with linear regression
there could be multiple curves

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of this shape that could fit our data.

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So how do we find out
which one is the best one.

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Well we need to calculate the maximum
likelihood.

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And the way it's done
is by looking at each data point.

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So for example we start here
and finding out for the person of that age

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what would have this curve
that we're potentially considering.

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What what prediction would have it made.

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So this specific curve, 
this specific logistic regression

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that we've modeled, it
would have said that a person of that age

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has only a 3% chance

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of saying yes or taking up the offer.

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So it's a very low chance for,

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this person or this, a person of this age
to take up the offer.

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And then we would continue.

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So for the next point where what would
have the logistic regression model.

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So we know that this person took up
the offer.

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We know it's a
yes because that's our input data.

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But what would have
the logistic regression said

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if we didn't know that the person took up
the offer and the logistic regression?

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The point would be somewhere
here on the curve, and that would be 54%

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chance, for this person or for this data
point would be 92% for this data point,

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95% for this data point,
our logistic regression

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for that specific age,
a person of that age predicts 98%.

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So we've got all those numbers. Great.

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Now, just to make, to avoid cluttering
this image, we'll,

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do the bottom points on a separate image.

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So for this point, the logistic regression
predicts

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a 1% chance for this point
for somebody of that age.

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So logistic regression predicts
a 4% chance for the someone of that age.

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A 10% chance,

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for someone of that age of 58% chance,
and for someone of the age, a 96% chance.

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But remember, on the right side
over here with the nose,

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the values that we've illustrated
1%, 4%, 10%, 58% and 96%,

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those are the probabilities of somebody
saying, yes, taking up the offer.

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So the probabilities of them saying no
are one minus that value.

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So let's add that in there.

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So one minus each one of those values
is the probability of them saying no.

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So for example for the first point
it is actually a 99% chance.

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The logistic regression is predicting
that someone that of that age,

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there's a 99% chance
that they will say no to the offer.

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And now, we need to calculate likelihood.

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Likelihood is calculated
by simply multiplying all these numbers.

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So on the blue side
we multiply these numbers.

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On the red side we multiply these numbers.

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And we get our value for likelihood.

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And then the way to find the best fitting
curve is

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to look through
all possible sorts of curves.

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Of course, there's a more sophisticated
process behind this, but in a nutshell,

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you compare what the likelihoods
are of different curves.

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So let's say we our logistic regression
modeling process

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started with this curve and calculate
the likelihood to be this value.

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Then it went on to the next curve
and it calculated the likelihood

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to be this value. Then the next curve
the likelihood was this.

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And through this iterative process
it found the curve with a maximum

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likelihood.

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So this would be the maximum
likelihood of all the curves.

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And that means this is the best curve.

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And that's
how the maximum likelihood is calculated.

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And that's how the best fitting
curve of logistic regression is found.

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I look forward to seeing you next time.

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And until then, enjoy machine learning.