Suppose you need to find a function to classify cases into k > 2 multiple classes….

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Questions

1. Suppose you need to find a function to classify cases into k > 2 multiple classes. Use the following notation. Denote X the space of the input variable X, and Y = {1,2,…,14 the space of the output variable Y. Define lly(x) = P(Y = y IX = x). In words, ny(x) is the probability that Y belongs to the class y when X = x. Note: For this question, your effort will be considered. So, even if you don’t get to the final solution, make sure to show your steps. (a) (10 points) Show that if we select a 0— 1 loss function, the risk is equivalent to the uncon-ditional probability that y f (x). (b) (20 points) Find the best a classifier (called Bayes Classifier) f* : X -> 9. Consider best the classifier that minimizes the risk using the 0 —1 loss function. • Hint 1: You will see that the classifier must give f* = argminR(f) = argmaxiy(x) I yeY

(c) (20 points) Show that

vi. : x -> y,R(f) -R(f*) = E [max {P(Y =y I X)} — P(Y = f (X) I X)] ycY

Interpret the quantity R(f) —R(f*). What does it measure?

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