  Quantum metrology uses small changes in the output probabilities of a quantum
measurement to estimate the magnitude of a weak interaction with the system.
The sensitivity of this procedure depends on the relation between the input
state, the measurement results, and the generator observable describing the
effect of the weak interaction on the system. This is similar to the situation
in weak measurements, where the weak value of an observable exhibits a
symmetric dependence on initial and final conditions. In this paper, it is
shown that the phase sensitivity of a quantum measurement is in fact given by
the variance of the imaginary parts of the weak values of the generator over
the different measurement outcomes. It is then possible to include the
limitations of a specific quantum measurement in the uncertainty bound for
phase estimates by subtracting the variance of the real parts of the weak
values from the initial generator uncertainty. This uncertainty relation can be
interpreted as the time-symmetric formulation of the uncertainty limit of
quantum metrology, where the real parts of the weak values represent the
information about the generator observable in the final measurement result.
