I'm trying to understand a passage titled "Tree Representations for Study Designs" in these notes which says the following.
I understand the setup but not much else. For example, my understanding was that a prospective design could also be $E\to S\to F$, i.e. individuals are selected based on their exposure, and monitored to see if they develop the disease. And I thought that a retrospective study design would be $F\to S\to E$. I thought retrospective meant that inclusion in the study is based on the outcome $F$. I also don't understand how he interprets prospective/retrospective design in terms of $P(E\cap F\cap S)$.
Can someone explain this mathematical perspective of prospective vs retrospective designs or provide a reference that expands on these ideas?