  A number of authors have suggested that nonlinear interactions can enhance
resolution of phase shifts beyond the usual Heisenberg scaling of 1/n, where n
is a measure of resources such as the number of subsystems of the probe state
or the mean photon number of the probe state. These suggestions are based on
calculations of `local precision' for particular nonlinear schemes. However, we
show that there is no simple connection between the local precision and the
average estimation error for these schemes, leading to a scaling puzzle. This
puzzle is partially resolved by a careful analysis of iterative implementations
of the suggested nonlinear schemes. However, it is shown that the suggested
nonlinear schemes are still limited to an exponential scaling in \sqrt{n}.
(This scaling may be compared to the exponential scaling in n which is
achievable if multiple passes are allowed, even for linear schemes.) The
question of whether nonlinear schemes may have a scaling advantage in the
presence of loss is left open.
  Our results are based on a new bound for average estimation error that
depends on (i) an entropic measure of the degree to which the probe state can
encode a reference phase value, called the G-asymmetry, and (ii) any prior
information about the phase shift. This bound is asymptotically stronger than
bounds based on the variance of the phase shift generator. The G-asymmetry is
also shown to directly bound the average information gained per estimate. Our
results hold for any prior distribution of the shift parameter, and generalise
to estimates of any shift generated by an operator with discrete eigenvalues.
