  We analyze the problem of estimating past quantum states of a monitored
system from a mathematical perspective in order to ensure self-consistency with
the principle of quantum non-demolition. Despite several claims of ``measuring
noncommuting observables'' in the physics literature, we show that we are
always measuring commuting processes. Our main interest is in the notion of
quantum smoothing or retrodiction. In particular, we examine proposals to
estimate the result of an external measurement made on an open quantum systems
during a period where it is also undergoing continuous monitoring. A full
analysis shows that the non-demolition principle is not actually violated, and
so a well-posed as a statistical inference problem can be formulated. We extend
the formalism to consider multiple independent external measurements made on
the system over the course of a continual period of monitoring.
