Probability condiioned on two variables

Question

Given random variables $X, Y, Z$, when does $p(X|Y, Z) = p(X|Y)p(X|Z)$? Is such a transformation ever justified?

Answer 1

Almost never.

Let $p(x|y,z)=P(X=x\mid Y=y,Z=z)$, $q(x|y)=P(X=x\mid Y=y)$ and $r(x|z)=P(X=x\mid Z=z)$, then $\sum\limits_xq(x\mid y)=1$ and $r(x\mid z)\leqslant1$ for every $x$ hence $\sum\limits_xq(x\mid y)r(x\mid z)\leqslant1$ always and $\sum\limits_xq(x\mid y)r(x\mid z)\lt1$ most of the time.

On the other hand, $\sum\limits_xp(x\mid y,z)=1$ hence the identity $p(x|y,z)=q(x|y)r(x|z)$ is quite unlikely.

Exercise: Determine the degenerate cases when, nevertheless, the equality holds.