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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.
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