p(X|Y, Z) = p(Y|X, Z)/p(X|Z)/p(Y|Z)

Question

Can anyone help me understand what

$$p(X|Y, Z) = \frac{p(Y|X, Z)p(X|Z)}{p(Y|Z)}$$

means? Or how it's derived (in simple language).

I know the story behind

$$p(X, Y|Z) = p(X|Z)p(Y|X, Z)$$

and I think they're somewhat similar to eachother(intuitively) but I don't know how. Any help would be grateful.

Thank you.

Answer 1

For a probability space $(\Omega,\mathscr{F},\mathbb{P})$ and a non-null event $Z\in\mathscr{F}$, we have $\mathbb{P}_Z\colon X\in\mathscr{F}\mapsto\mathbb{P}(X\mid Z)$ is a probability on $(\Omega,\mathscr{F})$ (exercise). Thus your equation is just the familiar Bayes' theorem for $\mathbb{P}_Z$.