  Monitoring a quantum observable continuously in time produces a stochastic
measurement record that noisily tracks the observable. For a classical process
such noise may be reduced to recover an average signal by minimizing the mean
squared error between the noisy record and a smooth dynamical estimate. We show
that for a monitored qubit this usual procedure returns unusual results. While
the record seems centered on the expectation value of the observable during
causal generation, examining the collected past record reveals that it better
approximates a moving-mean Gaussian stochastic process centered at a distinct
(smoothed) observable estimate. We show that this shifted mean converges to the
real part of a generalized weak value in the time-continuous limit without
additional postselection. We verify that this smoothed estimate minimizes the
mean squared error even for individual measurement realizations. We go on to
show that if a second observable is weakly monitored concurrently, then that
second record is consistent with the smoothed estimate of the second observable
based solely on the information contained in the first observable record.
Moreover, we show that such a smoothed estimate made from incomplete
information can still outperform estimates made using full knowledge of the
causal quantum state.
