Please just help with Part b

[5pts] Optimal predictors for the 0-1 loss. In this question, we will study the 0-1 loss in more detail, and derive the optimal predictor for classification. We will consider a simple binary data problem. The input X 6 {0, 1} and label 7 6 {0, 1} are binary random variables, and the set of predictors that we consider are the functions y : {0, 1} - {0, 1}. Recall the 0-1 loss when predicting t with y(I), Lo-1(y(x), t) = 0 ify(x) = t 1 if y(x) # t' (0.2) and recall the definition of the expected error, Rly] = ELLo-(y(X), T)] = > > Lo-(y(x), t) P(X = z, T = t), (0.3) te(0,1} ze{0,1} For this problem, we will assume that P(X = x) > 0 for all r. We use the following short hand. p(x, t) := P(X = r, T = t) (0.4) p(t x) := P(T = t[X = 1) (0.5) (a) [1pt] Assuming that p(0|x) = p(1|x) = 1/2 for all x 6 {0, 1}, prove that all predictors y : {0, 1} - {0, 1} achieve Rly] = 1/2 (0.6) (b) [3pt] Assuming that p(0|z) # p(1|x) for all r e {0, 1}, prove that y*(x) = arg max p(tz) te(0,1) (0.7) is the unique optimal predictor. In other words, show that Rly*] S Ry] (0.8) with equality only if y*(x) = y(x) for all r e {0, 1}. (c) [1pt] Give an example of a joint distribution p(x, t) on {0, 1}2 such that there are exactly two distinct optimal predictors y, and y2