Gaussian Mixture Models for Clustering

Now that we provided some background on Gaussian distributions, we can turn to a very important special case of a mixture model, and one that we're going to emphasize quite a lot in this course and in the assignment, and that's called a mixture of Gaussians. And remember that for any one of our image categories, and for any dimension of our observed vector like the blue intensity in that image, we're going to assume a Gaussian distribution to model that random variable. So for example, for forest images, if we just look at the blue intensity, then we might have a Gaussian distribution shown with the green curve here, which is centered about this value 0.42. And I want to mention here that we're actually assuming a Gaussian for the entire three-dimensional vector RGB. And that Gaussian can have correlation structure and it will have correlation structure between these different intensities, because the amount of RGB in an image tends not to be independent, especially within a given image class. But for the sake of illustrations and keeping all the drawings simple, we're just going to look at one dimension like this blue intensity here. But really, in your head, imagine these Gaussians in this higher dimensional space.........