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I am going through ISL(Python) and in section 2.2.3 ( Page No. 36), the author writes,

"For our simulated data, the Bayes error is 0.133. It is greater than zero, because the classes overlap in the true population... The Bayes error rate is analogous to the irreducible error, discussed earlier."

I am not able to figure out how the Bayes error is been calculated ? Suppose we have a simulation function f(x) = Sin(x) and a point is labelled as Y = 1 ;if f(x) >= 0 and Y = 0 ; otherwise

How should we approach to calculate the Bayes error? Also if you could please elaborate on how and why Bayes error rate is equivalent to irreducible error in regression setting.

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