For a Bayes Classifier are actual outcome and classifier outcome independent?

Question

For a Bayes Classifier defined as: $$ h(x) = \begin{cases} 1: Pr(Y=1 \mid X=x) > \eta \\ 0: \text{Otherwise} \end{cases} $$

Is this true: $$ Pr(Y=i, h(x) = j \mid X=x) = Pr(Y=i \mid X=x) * Pr(h(x) = j \mid X=x) $$ That is are they conditionally independent?

Answer 1

$h$ is constant given $X$. A constant random variable is independent of all random variables. Thus $h$ is independent of $Y$ given $X$.