[MUSIC] But before we get to our mixture model, I
want to provide some background on one of the components that's going to be a really
really key component to the model that we're going to describe, and this is
something called a Gaussian distribution. So, let's go back to this histogram
over blue intensities just for the cloud images, and we said this histogram might look
something like this bell-shaped curve. Well, when we're taking our probabilistic
model-based approach, we're going to treat the blue intensity in every image
as an observed random quantity. And we're going to place a distribution
over that random quantity, and that distribution is going to
have a set of perimeters, and we are going to aim to learn
those perimeters from the data. And so the distribution that
we are going to use here, to model this type of shape,
this type of spread of data points, is something called
a Gaussian distribution. And in this application,
we're going to assume that a Gaussian distribution provides
a pretty good fit for every different image category,
like clouds, sunsets, and forests. And every dimension of the observed
vector, whether we're looking at the red, green, or blue dimension. So for example, when we're just looking
at our blue intensity dimension, then this Gaussian is fully specified by
two parameters, a mean and a variance. Or sometimes people refer instead
to the square root of the variance, which is called the standard deviation. So the mean specifies where this Gaussian
lives, so it centers the distribution, and the variance determines
the spread of the distribution. So for example, here's a Gaussian
with the same variance, but a different mean, so a smaller mean
than the example we showed before. And now here's a Gaussian
with the same mean, but smaller variance, and here examples
with larger and larger variances. So we can see how tuning these
two different parameters changes what this distribution looks like,
in terms of its location and its spread. And we're going to notate this Gaussian
distribution as follows with this N, sometimes the Gaussian distribution is
referred to as the normal distribution, that's where the N comes in. And this X is going to refer to the random variable over which
this distribution is placed. So, in the example we're looking at,
that's the blue intensity in the image. And then, everything to the right
of the bar represents the fixed parameters of the Gaussian,
the mean and the variance. [MUSIC]