  In weak gravitational lensing, weighted quadrupole moments of the brightness
profile in galaxy images are a common way to estimate gravitational shear. We
employ general adaptive moments (GLAM) to study causes of shear bias on a
fundamental level and for a practical definition of an image ellipticity. The
GLAM ellipticity has useful properties for any chosen weight profile: the
weighted ellipticity is identical to that of isophotes of elliptical images,
and in absence of noise and pixellation it is always an unbiased estimator of
reduced shear. We show that moment-based techniques, adaptive or unweighted,
are similar to a model-based approach in the sense that they can be seen as
imperfect fit of an elliptical profile to the image. Due to residuals in the
fit, moment-based estimates of ellipticities are prone to underfitting bias
when inferred from observed images. The estimation is fundamentally limited
mainly by pixellation which destroys information on the original, pre-seeing
image. We give an optimized estimator for the pre-seeing GLAM ellipticity and
quantify its bias for noise-free images. To deal with pixel noise, we consider
a Bayesian approach where the posterior of the GLAM ellipticity can be
inconsistent with the true ellipticity if we do not properly account for our
ignorance about fit residuals. This underfitting bias is S/N-independent but
changes with the pre-seeing brightness profile and the correlation or
heterogeneity of pixel noise over the post-seeing image. Furthermore, when
inferring a constant ellipticity or, more relevantly, constant shear from a
source sample with a distribution of intrinsic properties (sizes, centroid
positions, intrinsic shapes), an additional, now noise-dependent bias arises
towards low S/N if incorrect priors for the intrinsic properties are used. We
discuss the origin of this prior bias.
