4.7 (Bayes decision theory) Consider a binary classification task, in which the label y{1,2}. If an example xR belongs to class 1 , it is generated by the class conditional p.d.f. p(xy=1)=N(1,0.25), and a class 2 example is sampled from the class conditional distribution p(xy=2)=N(1,0.25). Suppose Pr(y=1)= Pr(y=2)=0.5. (a) What is the p.d.f. p(x) ? (b) Let us use the cost matrix [0110]. Show that for any x, if we choose f(x)= argmaxyp(yx) to be our prediction for x, the cost E(x,y)[cy,f(x)] is minimized, and hence it is the optimal solution. Is this rule optimal if y{1,2,,C}(C>2) (i.e., in a multiclass classification problem)? (c) Using the cost matrix [0110] and Bayes decision theory, which classification strategy will be optimal for this task? What is the Bayes risk in this example? (d) If the cost matrix is [01100] (i.e., when the true label is 1 but the prediction is 2 , the cost is increased to 10 ), what is the new decision rule