[willdearden]

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.