  We show that postselection offers a nonclassical advantage in metrology. In
every parameter-estimation experiment, the final measurement or the
postprocessing incurs some cost. Postselection can improve the rate of Fisher
information (the average information learned about an unknown parameter from an
experimental trial) to cost. This improvement, we show, stems from the
negativity of a quasiprobability distribution, a quantum extension of a
probability distribution. In a classical theory, in which all observables
commute, our quasiprobability distribution can be expressed as real and
nonnegative. In a quantum-mechanically noncommuting theory, nonclassicality
manifests in negative or nonreal quasiprobabilities. The distribution's
nonclassically negative values enable postselected experiments to outperform
even postselection-free experiments whose input states and final measurements
are optimized: Postselected quantum experiments can yield anomalously large
information-cost rates. We prove that this advantage is genuinely nonclassical:
no classically commuting theory can describe any quantum experiment that
delivers an anomalously large Fisher information. Finally, we outline a
preparation-and-postselection procedure that can yield an arbitrarily large
Fisher information. Our results establish the nonclassicality of a metrological
advantage, leveraging our quasiprobability distribution as a mathematical tool.
