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[MUSIC]

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At the very core, of, measuring how
sure we are about a prediction,

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is a notion of probability.

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So let me do a very, very, very, very,
very quick review, of probabilities here.

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And how they're useful throughout this
module, and throughout the course.

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Okay.

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Now we'll see a very quick
review of probability,

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which is just showing a few examples,
and interpreting what probability means.

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So, if I say the probability that
a review is positive 0.7, so

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that means like in general,

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a 0.7 is a probability associated
with people writing positive reviews.

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What does that mean?

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Well if I take my data set of reviews,
I have to expect on

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average that 70% of the rows here,
will have positive reviews.

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And the other 30%, will be negative.

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Of course,
a data set is kind of a finite sample or

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observation of the underlying
space of reviews, so

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its not going to be exactly 70% but
its about 70%.

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That's how we interpret probabilities.

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Now we can associate probabilities with
what's called degrees of beliefs or

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degrees of sureness.

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So for example, let's look at
the probability that y is plus 1.

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So this is the notation that we'll use,
so probability.

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That y, which is the thing
that we're trying to predict,

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output is positive, and
let's interpret that.

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That output can range from 0 to 1.

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So if the output is 0 that
means that I'm absolutely sure

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that every single review
in the world is positive.

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That's what it means.

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So what this is saying,
is that the probability that

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y = +1 is equal to 1,
what does that imply?

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Well we have the probability that
y is -1 that reviews are negative

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is 1 minus the probability
that reviews are positive.

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Y = + 1, which means that it is 0.

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It means that there's 0 chance that there
are negative reviews, which is not true.

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And so, on the other hand, if I say that
the probability of y equals plus 1 is 0.

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That means I'm absolutely sure that every
review in the world is not positive.

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Which in our case would say that
the probability that y = + 1 is zero.

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And that implies that
the probability that y is -1 is one.

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That means that every review
out there is negative.

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And the truth is not that
somewhere in between.

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So for example,
if you say the probability is 0.5.

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That means that I'm not sure if
reviews are positive or negative.

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In general, they can be 50/50,
say on average half are positive,

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half are negative.

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And so this would say that
the probability that y is equal plus 1

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is equal to the probability that y
is equal to minus 1, which is 0.5.

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In other words,
the world is fair and balanced.

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50-50.

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So let's discuss some fundamental
properties of probabilities which I've

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hinted at in the examples before.

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So first of all, probabilities
are always between zero and one.

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So there are no negative probabilities.

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In fact,
I've taught classes where I've had

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students submit assignments that
had negative probabilities in them.

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Not true, cannot happen.

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So the probability that y is + 1

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is somewhere between 0 and 1.

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Similarly, probability that Y
is -1 is always greater than or

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equal to 0, and
is always less than equal to 1.

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Fundamental property.

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The other fundamental
property of probability is,

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is that probabilities add up to one.

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So the probability that y = +1 plus

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the probability that y = -1.

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These are two possibilities that
use either positive or negative,

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nothing else can happen
than that adds up to 1.

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So two things to remind ourselves.

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Now we just talked about the binary
classification case, but

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you can have multiple classes.

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So for example if it's an image maybe
it can be either dogs, cats, or birds.

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This is the only three things can
happen in the world, let's say.

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And in that case we have the probability,

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the three probabilities
are the property of y is a dog,

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so the image of the dog,
the probability that y is a cat,

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and the probability that y is a bird.

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And we have that all of
these are between 0 and 1.

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So they're all greater than 0 and
less than 1.

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But that's not enough to
capture what happens.

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The last important part is that
these probabilities add up to 1.

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That is, if that's the only thing that
images can be, dogs, cats, and birds.

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If the other things could be then
you have to add all those in.

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But in our case there's only
three possibilities, so

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we had the probability that y equals dog,
plus the probability that

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y is equal to cat,
plus probability that y is equal

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to bird, is equal to 1.

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And there you go.

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Now you know everything you ever
need to know about probabilities.

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That's it.

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[MUSIC]