Consider a pattern recognition problem with prior probabilities P (0) = 0.2 and P (1) = 0.8 and conditional distributions P (x|0) and P (x|1). Let R∗ denote the Bayes error rate for this problem. As usual, suppose we have independent and identically distributed training data (x1, 1), . . . , (xn, n). Let M be some universally consistent decision rule for this problem.

(a) Suppose we randomly select half of the training points and throw them away. Will the remaining training points be independent? Identically distributed?

(b) If we use method M but on the training data as modified in part (a), what asymptotic error rate will we get?

(c) Now suppose we take the original training data and throw away all the examples that have a label