  The observables of a noisy quantum system can be estimated by appropriately
filtering the records of their continuous measurement. Such filtering is
relevant for state estimation and measurement-based quantum feedback control.
It is therefore imperative that the observables estimated through a causal
filter satisfy the Heisenberg uncertainty principle. In the Markovian setting,
prior work implicitly guarantees this requirement. We show that any causal
estimate of linear observables of a linear, but not necessarily Markovian,
system will satisfy the uncertainty principle. In particular, this is true
irrespective of any feedback control of the system and of where in the feedback
loop -- inside or outside -- the measurement record is accessed. Indeed, causal
estimators using the in-loop measurement record can be as precise as those
using the out-of-loop record. These results clarify the role of causal
estimators to a large class of quantum systems, restores the equanimity of
in-loop and out-of-loop measurements in their estimation and control, and
simplifies future experiments on measurement-based quantum feedback control.
