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From this link, where the statement is given for events and not random variables, I gather that for random variables $X,Y,Z$, $P(Y|X)=P(Y|X,Z)$ only if $P(Y,Z|X)=P(Y|X)P(Z|X)$? Does this imply that $Y$ and $Z$ being conditionally independent (conditioned on $X$) is sufficient for $P(Y|X)=P(Y|X,Z)$ to hold? Further conditions on the relationship between $X,Y,Z$ (such as independence between any of the two or mutual independence of all 3) is not required?

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No, it's not required. Being $Y$ and $Z$ conditionally independent on $X$ directly implies $$P(Y|X)=P(Y|X,Z)$$

because $$P(Y|X,Z)=\frac{P(Y,Z|X)}{P(Z|X)}=\frac{\overbrace{P(Y|X)P(Z|X)}^{\text{conditional indep.}}}{P(Z|X)}=P(Y|X)$$

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