(b) Choose the best polynomial model obtained from the previous part, and use to it regress the entire dataset. Report the polynomial coefficients and make a scatter plot of the my's and ye's with your fitted polynomial. Problem 3. Bayes Optimal us. Logistic Regression. Recall that in classification, we assume that each data point (2;, y;) is drawn i.i.d. from a joint distribution P, i.e. P(X = x, Y = y) = P(Y = y)P(X = =\\Y = y)). In this problem, we will examine a particular distribution on which Logistic Regression is optimal. Suppose that this distribution is supported on rER,y ( {-1, +1}, and given by: P(Y = +1) = P(Y = -1) =1/2 P(X = x|Y = +1) = 1 - (1 5)2 V27T 1 (2+5)2 P(X = x|Y = -1) = V2T (a) Show that the joint data distribution is given by P(X = x, Y = y) = 1 (x-5y)2 (b) Plot (either using code or hand-drawn neatly) the conditional distributions P(X = x Y = +1) and P(X = x|Y = -1) in a single figure. Note: these are just Gaussian PDFS. (c) Write the Bayes optimal classifier h*(x) given the above distribution P and simplify. Hint: you should get a classification rule that classifies z based on whether or not it is greater than a threshold. (d) Compute the classification error rate of the Bayes optimal classifier, i.e. Pr (h*(x) / y) = E liot] Hint: Your result should be of the form 1 - (c), where d(.) is the Gaussian CDF. (e) (Extra Credit) Recall that Logistic Regression assumes that the data distribution is of the form P(Y = +1[X = x) = 1 te Po-Biz Show that the distribution given above satisfies this assumption. What values of Do, B1 does this correspond to