Suppose $ X$ is discrete and $ Z,W$ are discrete or continuous, I am wondering if it is always the case (or at least non-trivially) that
$ $ P(X=x\mid Z) \geq P(X=x\mid Z,W) $ $
for all $ x\in X$ .
It appears to make intuitive sense to me because it appears the probability of $ X$ given $ Z$ and $ W$ should be “subsetting” off the probability of $ X$ given $ Z$ . That is, if there is some chance of $ X$ given $ Z$ , then $ X$ given $ Z$ and $ W$ is subdividing the occurrence of $ X$ given $ Z$ into many more chunks according to $ W$ , and thus the probability of a chunk occurring should be less than the whole.
In other words, it seems to hold in the finite-sampling perspective, where the above are empirical proportions over some collection of objects. However, the result doesn’t seem to hold more generally. I am wondering why the finite-sampling intuition doesn’t seem to extend. Is there a general result behind this?