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In the setting of this problem, $\eta(\vec{x})$ is $P(Y=1|\vec{X}=\vec{x})$, $Y \in {0,1}$, $X \in R^d$. Being the true probability know, the classification rule is simply $\eta(\vec{x})>0.5 \Rightarrow \hat{Y}=1$. The risk of misclassification is, as usual, $E[min(\eta(\vec{x}),1-\eta(\vec{x}))]$. However, I am asked to prove that if the Xs are iid conditional con Y, $X_i|Y \sim p(X|Y) \forall X_i$, then $$R^*_d \leqslant e^{-cd}$$ I have tried anything I could think of but I achieved very little. I see the intuition (if the features are all conditionally independent, increasing them add new information to the estiamtion and allows for a lower minimal error), but I have no idea of how to prove it formally. Big thanks in advance to anybody who is able to crack it!

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