Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 6 / a

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- Solution a

Solution

The binary regression functions are the conditional class probabilities

η_{1} (x) = P (Y = 1 ∣ X = x) = E (Y ∣ X = x), η_{0} (x) = 1 - η_{1} (x) .

Under zero-one loss, the Bayes classifier is

C^{*} (x) = 1_{{η_{1} (x) \geq 1/2}},

with arbitrary tie breaking. Its Bayes decision boundary is

{x : η_{1} (x) = η_{0} (x)} = {x : η_{1} (x) = 1/2} .

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