  We study the problem of estimating the magnitude of a Gaussian beam
displacement using a two pixel or 'split' detector. We calculate the maximum
likelihood estimator, and compute its asymptotic mean-squared-error via the
Fisher information. Although the signal-to-noise ratio is known to be simply
related to the Fisher information under idealised detection, we find the two
measures of precision differ markedly for a split detector. We show that a
greater signal-to-noise ratio 'before' the detector leads to a greater
information penalty, unless adaptive realignment is used. We find that with an
initially balanced split detector, tuning the normalised difference in counts
to 0.884753... gives the highest posterior Fisher information, and that this
provides an improvement by at least a factor of about 2.5 over operating in the
usual linear regime. We discuss the implications for weak-value amplification,
a popular probabilistic signal amplification technique.
