3 Evaluating Classifiers Consider the following decision rule ft for a two-category problem in R. Given an input x ER decide category y, if x > t; otherwise decide category y2 (1) What is the error rate for this rule, that is, what is P[fe(x) + y)? (ii) Show that at for the optimally selected threshold value t (i.e., the one which gives minimum error rate), it must be the case that P(X = t|Y = y)P(Y = y) = P(X = t|Y = y2)P(Y = y2). (iii) Assume that the underlying population distribution has equal class priors (i.e., PY = y] = PY = y2]), and the individual class conditionals (i.e., P[X|Y = y) and P[X|Y = yl) are distributed as Gaussians. Give an example setting of the class conditionals (i.e., give an example parameter settings for the Gaussians) such that for some threshold value t, the rule ft achieves the Bayes error rate; and similarly, give an example setting of the class conditionals such that for no threshold value t, the rule ft achieves the Bayes error rate. 3 Evaluating Classifiers Consider the following decision rule ft for a two-category problem in R. Given an input x ER decide category y, if x > t; otherwise decide category y2 (1) What is the error rate for this rule, that is, what is P[fe(x) + y)? (ii) Show that at for the optimally selected threshold value t (i.e., the one which gives minimum error rate), it must be the case that P(X = t|Y = y)P(Y = y) = P(X = t|Y = y2)P(Y = y2). (iii) Assume that the underlying population distribution has equal class priors (i.e., PY = y] = PY = y2]), and the individual class conditionals (i.e., P[X|Y = y) and P[X|Y = yl) are distributed as Gaussians. Give an example setting of the class conditionals (i.e., give an example parameter settings for the Gaussians) such that for some threshold value t, the rule ft achieves the Bayes error rate; and similarly, give an example setting of the class conditionals such that for no threshold value t, the rule ft achieves the Bayes error rate