What's the between $P(x|y)P(y|z)$ and $P(x|y,z)P(y|z)$, it seems to me they both equal to $P(x,y|z)$.
Does any condition should be satisfied if they both equal to $P(x,y|z)$.
Conditional probability and Bayes’ rule
bayes-theoremconditional probabilityprobability
Best Answer
By definition $\mathsf P(x,y\mid z)~=~\mathsf P(x\mid y,z)~\mathsf P(y\mid z)$ always holds for any random variables $x,y,z$ where $\mathsf P(y\mid z)\neq 0$.
Yes. $\mathsf P(x\mid y,z)=\mathsf P(x\mid y)$ exactly when $x\perp z\mid y$ (that is, $x$ and $z$ are conditionally independent when given $y$).
Only when one this is so can we make the substitution to claim $\mathsf P(x,y\mid z)~=~\mathsf P(x\mid y)~\mathsf P(y\mid z)$.