We will follow the notation in the lectures. Assume that Y takes values in {0, 1}. Based on the lectures, we know the Bayes classifier is f*(x) = 1 if pi(x) = P(Y = 1/X = x) > 1/2 and f*(x) = 0 otherwise. (I use a slightly different notation f* to denote the Bayes classifier rather than f in the slides). Since p1(x) depends on the unknown data distribution, the Bayes classifier is not implement table in practice. One way to construct a practical classifier is the following. Let us first use some statistical model to estimate pi(x). We call this estimator as pi(x) E [0, 1]. Then we can plug-in the Bayes classifier, which means replace pi(x) with p1(x). So, we have the following classifier f(x) = 1 if p1(x) > 1/2 and f(x) = 0 otherwise. Now, we define the excess risk of the classifier f(x) as R(f) - R(f* ), where R(f) = P(Y # f(X)) is the misclassification error of f (unconditioning on X). In words, the excess risk is the difference between the misclassification errors of f and the Bayes classifier. Since the Bayes classifier has the smallest misclassification error (shown in the class), we know the excess risk is always nonnegative. We can claim that f is a good classifier, if its excess risk is close to 0. So, for any given classifier f, we would like to know its excess risk or its upper bound. Given the above background, please prove the following inequality regarding the excess risk R(f) - R(f*)