Predictive approach to some quantum paradoxes

In classical probability theory, the best predictor of a future observation of a random variable $X,$ is its expected value $E_P[X]$ when no other information is available When information consisting in the observation of another random variable $Y$ is available, then the best predictor of $X$ is another random variable $E_P[X|Y].$ It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. We shall see that exactly as in classical prediction theory, when the result of an observation is taken into account by means of a non-commutative conditional expectation, some of the usual paradoxes cease to be such.