  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.
