# Bayesian error in binary classification when covariates are conditionally iid

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!