  Probabilistic metrology attempts to improve parameter estimation by
occasionally reporting an excellent estimate and the rest of the time either
guessing or doing nothing at all. Here we show that probabilistic metrology can
never improve quantum limits on estimation of a single parameter, both on
average and asymptotically in number of trials, if performance is judged
relative to mean-square estimation error. We extend the result by showing that
for a finite number of trials, the probability of obtaining better estimates
using probabilistic metrology, as measured by mean-square error, decreases
exponentially with the number of trials. To be tight, the performance bounds we
derive require that likelihood functions be approximately normal, which in turn
depends on how rapidly specific distributions converge to a normal distribution
with number of trials.
