[MUSIC] At the very core, of, measuring how
sure we are about a prediction, is a notion of probability. So let me do a very, very, very, very,
very quick review, of probabilities here. And how they're useful throughout this
module, and throughout the course. Okay. Now we'll see a very quick
review of probability, which is just showing a few examples,
and interpreting what probability means. So, if I say the probability that
a review is positive 0.7, so that means like in general, a 0.7 is a probability associated
with people writing positive reviews. What does that mean? Well if I take my data set of reviews,
I have to expect on average that 70% of the rows here,
will have positive reviews. And the other 30%, will be negative. Of course,
a data set is kind of a finite sample or observation of the underlying
space of reviews, so its not going to be exactly 70% but
its about 70%. That's how we interpret probabilities. Now we can associate probabilities with
what's called degrees of beliefs or degrees of sureness. So for example, let's look at
the probability that y is plus 1. So this is the notation that we'll use,
so probability. That y, which is the thing
that we're trying to predict, output is positive, and
let's interpret that. That output can range from 0 to 1. So if the output is 0 that
means that I'm absolutely sure that every single review
in the world is positive. That's what it means. So what this is saying,
is that the probability that y = +1 is equal to 1,
what does that imply? Well we have the probability that
y is -1 that reviews are negative is 1 minus the probability
that reviews are positive. Y = + 1, which means that it is 0. It means that there's 0 chance that there
are negative reviews, which is not true. And so, on the other hand, if I say that
the probability of y equals plus 1 is 0. That means I'm absolutely sure that every
review in the world is not positive. Which in our case would say that
the probability that y = + 1 is zero. And that implies that
the probability that y is -1 is one. That means that every review
out there is negative. And the truth is not that
somewhere in between. So for example,
if you say the probability is 0.5. That means that I'm not sure if
reviews are positive or negative. In general, they can be 50/50,
say on average half are positive, half are negative. And so this would say that
the probability that y is equal plus 1 is equal to the probability that y
is equal to minus 1, which is 0.5. In other words,
the world is fair and balanced. 50-50. So let's discuss some fundamental
properties of probabilities which I've hinted at in the examples before. So first of all, probabilities
are always between zero and one. So there are no negative probabilities. In fact,
I've taught classes where I've had students submit assignments that
had negative probabilities in them. Not true, cannot happen. So the probability that y is + 1 is somewhere between 0 and 1. Similarly, probability that Y
is -1 is always greater than or equal to 0, and
is always less than equal to 1. Fundamental property. The other fundamental
property of probability is, is that probabilities add up to one. So the probability that y = +1 plus the probability that y = -1. These are two possibilities that
use either positive or negative, nothing else can happen
than that adds up to 1. So two things to remind ourselves. Now we just talked about the binary
classification case, but you can have multiple classes. So for example if it's an image maybe
it can be either dogs, cats, or birds. This is the only three things can
happen in the world, let's say. And in that case we have the probability, the three probabilities
are the property of y is a dog, so the image of the dog,
the probability that y is a cat, and the probability that y is a bird. And we have that all of
these are between 0 and 1. So they're all greater than 0 and
less than 1. But that's not enough to
capture what happens. The last important part is that
these probabilities add up to 1. That is, if that's the only thing that
images can be, dogs, cats, and birds. If the other things could be then
you have to add all those in. But in our case there's only
three possibilities, so we had the probability that y equals dog,
plus the probability that y is equal to cat,
plus probability that y is equal to bird, is equal to 1. And there you go. Now you know everything you ever
need to know about probabilities. That's it. [MUSIC]