tokens: 5 . Nevertheless, the resulting statistical errors of shear, as opposed to its bias, are larger by roughly the factor 1.7 with the uniform prior, which reflects the larger a-priori uncertainty on . This implies that a shear bias due to an incorrect prior is negligible in this experiment. 5. Discussion Unlike galaxy ellipticities defined in terms of unweighted moments, GLAM provide a practical definition of ellipticity with useful properties for any chosen adaptive weight : they (i) are identical with the ellipticity of isophotes of elliptical galaxy images; (ii) under the influence of shear they behave exactly like defined with unweighted moments; (iii) they are unbiased estimators of reduced shear; (i)(iii) assume ideal conditions without pixellation, PSF convolution, or pixel noise. These effects fundamentally limit our ability to determine the GLAM ellipticity (see below). Under ideal conditions, see SS97, it is known that for unweighted moments already obeys (iii). However, unweighted moments are formally infinite for realistic galaxy images because of pixel noise so that weighting is a necessity which is done adaptively for GLAM . For adaptive weights, we have shown (i) and (ii) in Sects. 2.2 and 2.3 . Their relevance as unbiased estimators for reduced shear, statement (iii), follows thereafter from (ii) and the conclusions in SS97 for unweighted moments. We emphasise that GLAM do not require “deweighting” in a lensing analysis ( Melchior et al. 2011 ): the ellipticity in term of the weighted moments is actually unbiased. In particular, we do not have to devise the same weight function for all galaxies as any weight function equally obeys (ii). If the minimum of the functional ( 3 ) uniquely exists, the moments of the best-fitting template will, for a constant scalar , be the adaptively weighted moments of the galaxy light profile for the weight (Appendix A ). This means that the radial profile of the weight at separation from the centroid is for a template profile . We exploited this relation between adaptive moments and moments of a best-fitting elliptical profile to analyse limits to adaptive moment-based methods under the loss of image information. Namely, for realistic images subject to pixellation and PSF effects, underfitting bias can be present for GLAM if estimated from the post-seeing image. It potentially arises because of a loss of information on the pre-seeing galaxy image, fundamentally due to pixellation. To explore the limitations, we assume noise-free images and express in Sect. 2.4 the mapping between a pre-seeing and post-seeing image by the linear operation . The pre-seeing adaptive moments are equivalent to a least-square fit with residual of an elliptical profile to the pre-seeing image . If estimated from the post-seeing image whilst ignoring residuals, the inferred ellipticity is biased for and a singular . The latter indicates a loss of information on the pre-seeing image. For noise-free images and the estimator
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