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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.
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