The binary <regression function>[regression functions] are the <conditional class probabilities>
$$
\eta_1(x)=\mathbb P(Y=1\mid X=x)=\mathbb E(Y\mid X=x),
\qquad
\eta_0(x)=1-\eta_1(x).
$$
Under zero-one loss, the <Bayes classifier> is
$$
C^*(x)=\mathbf1_{\{\eta_1(x)\geq1/2\}},
$$
with arbitrary tie breaking. Its <Bayes decision boundary> is
$$
\{x:\eta_1(x)=\eta_0(x)\}
=\{x:\eta_1(x)=1/2\}.
$$

Solution

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

{title2=$m(x)=\mathbb E[Y\mid X=x]$}

A regression function is a conditional expected response. For binary labels it equals the conditional probability of class one.


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