Question 2 (50 '55) A 3dimensional random vector X takes values from a mixture of four Gaussians. One of these Gaussians represent the classconditional pdf for class 1, and another Gaussian represents the classconditional pdf for class 2. lE'lass 3 data originates from a mixture of the remaining 2 Gaussian components with equal weights. For this setting where labels L E {1,2, 3}, pick your own classconditional pdfs p{x|L = j}, j E {1,2, 3} as described. Try to approximately set the distances behveen means of pairs of Gaussians to twice the average standard deviation of the Gaussian compoenents, so that there is some signicant overlap between classconditional pdfs. Set class priors to 13.3, ID. 3, 13.4. Part A: Minirnum probability of enor classication [l loss, also referred to as Bayes Deci sion rule or MAP classier}. 1. Generate If!!!) samples from this data distribution and keep track of the true labels of each sample. 2. Specify the decision rule that achieves minimum probability of error {i.e., use {1-1 loss}, implement this classier with the true data distribution knowledge, classify the lK samples and count the samples corresponding to each decisionlabel pair to empirically estimate the confusion matrix whose entries are HI) = i|L = j} for i, j E {1,2, 3}. 3. Provide a visualization of the data {scatter-plot in 3-dimensional space}, and for each sample indicate the true class label with a different marker shape (dot, circle, triangle, square} and whether it was correctly {green} or incorrectly {red} classied with a different marker color as indicated in parentheses. Part B: Repeat the exercise for the ERM classication rule with the following loss matrices which respectively care If) times or l times more about not making mistakes when L = 3: 1 If) [I l IUD hm = 1 {1' If) and lm = 1 IUD {I} l 1 1 1 {1' Note that, the (i, j)\" entry of the loss matrix indicates the loss incurred by deciding on class i when the true label is j. For this part, using the lK samples, estimate the minimum expected risk that this optimal ERM classication rule will achieve. Present your results with visual and numerical reprentations. Briey discuss interesting insights, if any