  We investigate the problem of noise bias in maximum likelihood and maximum a
posteriori estimators for cosmic shear. We derive the leading and
next-to-leading order biases and compute them in the context of galaxy
ellipticity measurements, extending previous work on maximum likelihood
inference for weak lensing. We show that a large part of the bias on these
point estimators can be removed using information already contained in the
likelihood when a galaxy model is specified, without the need for external
calibration. We test these bias-corrected estimators on simulated galaxy images
similar to those expected from planned space-based weak lensing surveys, with
promising results. We find that the introduction of an intrinsic shape prior
can help with mitigation of noise bias, such that the maximum a posteriori
estimate can be made less biased than the maximum likelihood estimate.
Second-order terms offer a check on the convergence of the estimators, but are
largely sub-dominant. We show how biases propagate to shear estimates,
demonstrating in our simple setup that shear biases can be reduced by orders of
magnitude and potentially to within the requirements of planned space-based
surveys at mild signal-to-noise. We find that second-order terms can exhibit
significant cancellations at low signal-to-noise when Gaussian noise is
assumed, which has implications for inferring the performance of
shear-measurement algorithms from simplified simulations. We discuss the
viability of our point estimators as tools for lensing inference, arguing that
they allow for the robust measurement of ellipticity and shear.
