Please observe that \(p_{in}(t_{in}|t)\) is a so called backward probability, since it does not depend upon any future time. It can be constructed by available information about inputs and discharges only assuming to know everything necessary since \(t=-\infty\). This is obviously wishful thinking, because we have usually knowledge of the past up to some initial time \(t_0\) and not before.
Niemi's identity
Niemi's relation follows equating results from equation (4) and (14)