$X, Y$ and $Z$ are all random variables.
By definition, $P(X=x|Z=z)=P(X=x)$ implies that $X \perp Z$.
Does $P(X=x|Y=y,Z=z)=P(X=x|Y=y)$ also imply that $X \perp Z$?
2026-03-29 22:27:53.1774823273
Conditional Independence two random variables
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No. The random variables $X$ and $Z$ are independent conditionally on $Y$, which does not imply independence. The simplest counterexample is $X = X' + Y$ and $Z = Z' + Y$, where $X',Z',Y$ are independent $\mathrm{Bernoulli}(\frac12)$.