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I have been curious about this for a long time. Is there a sensible notion of a conditional random variable? I have been told "no" before, but I don't feel that I ever got a clear answer. And, in fact, I see them get used in statistics all the time. In particular, $x \sim (X|Y)$ is shorthand for

$$y\sim Y\\ x\sim (X|Y=y).$$

This "sample and bind" shorthand seems completely straight-forward to me, since we can "always" recover the distribution of $Y$ from the joint distribution of $(X,Y)$ by marginalizing $X$ out. Are marginalizing and conditioning adjoints? It seems that the notation witnesses triple-ability, if they are adjoint at all.

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    $\begingroup$ In terms of the probability behind this, you should look up disintegrations and regular conditional probabilities. I don't know about the categorial side of the question. $\endgroup$
    – Michael Greinecker
    Apr 29, 2014 at 16:50

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