This assignment is about fitting mixtures of axis-aligned Gaussians to two-dimensional data. Copy the files below into your directory: http://www.cs.toronto.edu/~hinton/csc321/matlab/mogem.m http://www.cs.toronto.edu/~hinton/csc321/matlab/moginit.m http://www.cs.toronto.edu/~hinton/csc321/matlab/plotellipse.m http://www.cs.toronto.edu/~hinton/csc321/matlab/showmog.m PART 1 (5 points) Run moginit to create training and validation datasets from 5 random axis aligned Gaussians. Then use the function mogem to fit various numbers of axis aligned Gaussians to the training data. Using performance on the validation data, determine the optimal number of Gaussians to fit to the training data. Present your results as a graph that plots both the validation density and the training density as a function of the number of Gaussians. Include a brief statement of what you think the graph shows. Also include a brief statement about the effects of changing the initial standard deviation used in mogem. Please do not change the random seeds in moginit.m (this will produce different data). You can do this later if you want to just play around and get a feel for how the algorithm behaves. PART 2 (2 points) Change moginit.m to use only 10 cases per Gaussian, and 10 axis-aligned gaussians to generate the data and repeat the experiment above (without changing the random seeds). Present your results as a graph and include a brief statement of what you think the graph shows and why it differs from the graph in PART 1. PART 3 (3 points) Change mogem.m so that in addition to fitting the means and axis-aligned variances, it also fits the mixing proportions. Currently, mogem does not mention mixing proportions so it is currently assuming that they are all equal (which makes them all cancel out when computing the posterior probability of each Gaussian for each datapoint.). So the first thing to do is to include mixing proportions when computing the posterior, but keep them fixed (and not all equal). Once you have debugged this, try learning the mixing proportions. Make mogem print out the final mixing proportions and hand in final mixing proportions that you get when fitting 4 Gaussians to the data, when you start with mixing proportions [0.25, 0.25, 0.25, 0.25], [0.3, 0.2, 0.2, 0.3], [0.2, 0.3, 0.3, 0.2] and [0.9, 0.025, 0.025, 0.025]. Do not change the data.