though in literature \(t_{ex}\) is usually labeled as \(t\).

Summary of probabilities

In our analysis we encountered several different probabilities of which we try a list here:
It is evident that all of these probabilities derive from the joint one, but under different operations. Niemi's identity relates \(p_{ex}\) and \(p_{in}\) .
One interesting goal is to find also a relation, for instance, between \(p_{ex}\left(t|t_{in} \right)\) and \(p(t_{ex}|t_{in})\) (and correspondingly between \(p_{in}(t_{in}|t)\) and \(p(t_{in}|t)\)). In Botter et al., 2011, this relation is set to be a function \(\omega(t,t_{in})\) for which: