There is a professor who was at Wisconsin, Charles Manski, who developed partial identification, which uses tons of these decompositions.

Idea is let's say you have a binary survey question where 80% respond and 90% of them respond "yes". What can we say about population "yes" rate (assume sample size is huge for simplicity)?

P(Yes) = P(Yes | response) * P(response) + P(Yes | no response) * P(no response) = 0.9 * 0.8 + P(Yes | no response) * 0.2 = 0.72 + P(Yes | no response) * 0.2

Then 0 <= P(Yes | no response) <= 1, so 0.72 <= P(Yes) <= 0.92. This example is somewhat trivial but it's a useful technique for showing exactly how your assumptions map to inferences.

For those who are wondering how this equation is true:

https://en.wikipedia.org/wiki/Law_of_total_variance

Confusing to me how Var[Y] can be computed only from terms that are conditional on X ... surely its marginal variance has components arising from variance outside its distribution that is conditional on X? (or is that a typo ....)

You can consider the tower rule for intuition on why this can be true:

    E[X] = E[E[X|Y]]

Essentially, what's happening is that, in E[X|Y], we first "hold" the randomness of Y, then consider the expected value of X for every possible value of Y. Once this is determined, we then take the expectation over all values of Y, which integrates out its effects, leaving us with the expected value of X.

This rule is similar, and arises from the tower rule above.

thanks! that kind of does make sense!

Similar to the law of total expectation, easier for me to think of partitioning a weighted mean :)