This is a made-up example, just to understand a concept. If changing the probability values aids your explanation, that's fine by me.
Say you have a physical quantity $E$ that can take values 1, 2, 3 with probabilities 0.4, 0.25, 0.35 respectively (working in a quantum framework). You have a positive operator valued measure, $E_1, E_2, E_3$, with $E_i$ corresponding to your measurement resulting in value $i$. If $\rho$ is the density operator representing the current state, then you have:
Tr($\rho E_1$) = 0.4, Tr($\rho E_2$) = 0.25, Tr($\rho E_3$) = 0.35
Given just these probability values, is it possible to construct $E_1, E_2, E_3$ "backwards"?