Issue 
A&A
Volume 604, August 2017



Article Number  A109  
Number of page(s)  18  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201629591  
Published online  22 August 2017 
Weaklensing shear estimates with general adaptive moments, and studies of bias by pixellation, PSF distortions, and noise
ArgelanderInstitut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: psimon@astro.unibonn.de
Received: 26 August 2016
Accepted: 18 May 2017
In weak gravitational lensing, weighted quadrupole moments of the brightness profile in galaxy images are a common way to estimate gravitational shear. We have employed 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 momentbased techniques, adaptive or unweighted, are similar to a modelbased 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, momentbased 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, preseeing image. We give an optimised estimator for the preseeing GLAM ellipticity and quantify its bias for noisefree images. To deal with images where pixel noise is prominent, we consider a Bayesian approach to infer GLAM ellipticity where, similar to the noisefree case, the ellipticity posterior can be inconsistent with the true ellipticity if we do not properly account for our ignorance about fit residuals. This underfitting bias, quantified in the paper, does not vary with the overall noise level but changes with the preseeing brightness profile and the correlation or heterogeneity of pixel noise over the 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 noisedependent bias arises towards low signaltonoise if incorrect prior densities for the intrinsic properties are used. We discuss the origin of this prior bias. With regard to a fullyBayesian lensing analysis, we point out that passing tests with source samples subject to constant shear may not be sufficient for an analysis of sources with varying shear.
Key words: gravitational lensing: weak / methods: data analysis / methods: statistical
© ESO, 2017
1. Introduction
Over the past decade measurements of distortions of galaxy images by the gravitational lensing effect have developed into an important, independent tool for cosmologists to study the largescale distribution of matter in the Universe and its expansion history (recent reviews: Schneider 2006; Munshi et al. 2008; Hoekstra & Jain 2008; Massey et al. 2010; Kilbinger 2015). These studies exploit the magnification and shear of galaxy light bundles by the tidal gravitational field of intervening matter. The shear gives rise to a detectable coherent distortion pattern in the observed galaxy shapes. The distortions are usually weak, only of the order of a few per cent of the unlensed shape of a typical galaxy image. Therefore, the key to successfully devising gravitational shear as cosmological tool is accurate measurements of the shapes of many, mostly faint and hardly resolved galaxy images.
There has been a boost of interest in methods of shape measurements in anticipation of the gravitational lensing analysis of the upcoming next generation of widefield imaging surveys (e.g. Euclid; Laureijs et al. 2011). Despite being sufficient for contemporary surveys, current methodologies are not quite at the required level of accuracy to fully do justice to the quantity of cosmological information we expect from future lensing surveys (Heymans et al. 2006; Massey et al. 2007). The challenge all methodologies face is that observable, noisy (postseeing) galaxy images are modifications of the actual (preseeing) image owing to instrumental and possible atmospheric effects. Postseeing galaxy images are subject to pixellation as well as instrumental noise, sky noise, photon noise, and random overlapping with very faint objects (Kitching et al. 2012; Hoekstra et al. 2015). In addition, galaxies are not intrinsically circular such that their ellipticities are noisy estimators of the cosmic distortion. Current theoretical work consequently focuses on sources of bias in shape measurements, such as pixelnoise bias, shapenoise bias, underfitting bias, colour gradients, or several selection biases (Hirata & Seljak 2003; Hirata et al. 2004; Mandelbaum et al. 2005; Melchior et al. 2010; Viola et al. 2011; Melchior & Viola 2012; Kacprzak et al. 2012; Massey et al. 2013; Semboloni et al. 2013).
One major source of bias is the pixelnoise bias or simply noise bias hereafter. This bias can at least partly be blamed on the usage of point estimates of galaxy shapes in a statistical analysis, that is singlevalue estimators of galaxy ellipticities (Refregier et al. 2012). This begs the question of whether it is feasible to eradicate noise bias by means of a more careful treatment of the statistical uncertainties in the measurement of galaxy ellipticities within a fully Bayesian framework. Indeed recent advances in image processing for weak gravitational lensing strongly support this idea, at least for the inference of constant shear (Sheldon 2014; Bernstein & Armstrong 2014, BA14 hereafter; Bernstein et al. 2016, BAKM16 hereafter). In contrast, the contemporary philosophy with point estimates is to perform elaborate, timeconsuming calibrations of biased estimators by means of simulated images; the calibration accuracy is, additionally, only as good as the realism of simulated images (e.g. Hoekstra et al. 2015). It is the case that code implementations of nonBayesian techniques are typically an order of magnitude or more faster than Bayesian codes which could be a decisive factor for upcoming surveys.
Here we take a new look into possible causes of bias in shear measurements on a fundamental level. To this end, we examine, stepbystep with increasing complexity, a fullyBayesian lensing analysis based on weighted brightness moments of galaxy images (Gelman et al. 2013; MacKay 2003). While the method in BA14 and BAKM16 is set in Fourier space, we work with moments in angular space which has benefits in the case of correlated noise or missing pixels in realistic images. Momentbased methods as ours are nonparametric; this means they are free from assumptions about the galaxy brightness profile. They hence appear to be advantageous for reducing bias, but nonetheless the specific choice of the adaptive weight for the moments is known to produce bias (Viola et al. 2014; Voigt & Bridle 2010). The origin of this problem, which principally also affects unweighted moments, becomes obvious in our formalism. We define as practical measure of galaxy shape a generalization of the impractical ellipticity ϵ expressed in terms of unweighted moments (Kaiser et al. 1995; Seitz & Schneider 1997, SS97 hereafter). Being Bayesian, our measurement of ellipticity results in a MonteCarlo sample of the probability distribution function (PDF) of ϵ which should be propagated in a fully Bayesian analysis. That is: we do not devise point estimators in order to ideally stay clear of noise bias. This overall approach of general adaptive moments (GLAM ) is inspired by and comparable to Bernstein & Jarvis (2002) apart from the Bayesian framework and some technical differences: (i) for any adaptive weight, the perfectly measured GLAM ellipticity is an unbiased estimator of gravitational shear unaffected by shapenoise bias; (ii) the adaptive weight may have a nonGaussian radial profile; (iii) our inference of the preseeing ellipticity is realised as forwardfitting of elliptical profiles (socalled templates), that is we do not determine the brightness moments of the postseeing image and correct them to estimate the preseeing moments (cf. Hirata & Seljak 2003; Mandelbaum et al. 2005).
As a disclaimer, the GLAM methodology outlined here is prone to bias, even where a fully Bayesian analysis can be realised, and, at this stage, its performance is behind that of other techniques. The aim of this paper is to elucidate causes of bias, instead of proposing a new technique that is competitive to existing techniques. However, these findings are also relevant for other methodologies because modelbased or momentbased approaches are linked to GLAM .
For this paper, we exclude bias from practically relevant factors: the insufficient knowledge of the PSF or noise properties, blending of images, and the selection of source galaxies (see e.g. Heymans et al. 2006; Hartlap et al. 2011; Dawson et al. 2016). We focus on the core of the problem of shape measurements which is the inference of preseeing ellipticities from images whose full information on the brightness profile have been lost by instrumental limitations.
The paper is laid out as follows. In Sect. 2, we introduce the formalism of GLAM for a practical definition of ellipticity with convenient transformation properties under the action of gravitational shear. We also analytically investigate the limits of measuring the preseeing ellipticity from a noisefree but both PSFconvolved and pixellated image. In Sect. 3, we construct a statistical model for the GLAM ellipticity of noisy postseeing images. We then study the impact of inconsistencies in the posterior model of ellipticity in three, increasingly complex scenarios. First, we analyse with independent exposures of the same preseeing image the ellipticity bias due to a misspecified likelihood in the posterior (underfitting bias). Second, we consider samples of noisy images with the same ellipticity but distributions of intrinsic properties such as sizes or centroid positions. Here a new contribution to the ellipticity bias emerges if the prior densities of intrinsic properties are incorrectly specified (prior bias). Third in Sect. 4, we perform numerical experiments with samples of noisy galaxy images of random intrinsic shapes that are subject to constant shear. With these samples we study the impact of inconsistent ellipticity posteriors on shear constraints (shear bias). We also outline details on our technique to MonteCarlo sample ellipticity or shear posteriors. We discuss our results and possible improvements of the GLAM approach in Sect. 5.
2. General adaptive moments
2.1. Definition
Let I(x) be the light distribution in a galaxy image of infinite resolution and without noise where x is the position on the sky. A common way of defining a (complex) ellipticity of a galaxy image uses the quadrupole moments, (1)of I(x) relative to a centroid position, (2)of the image (e.g. Bartelmann & Schneider 2001). For real applications, the quadrupole moments are soundly defined only if they involve a weight, decreasing with separation from the centroid position X_{0}, because galaxies are not isolated so that the normalisation and the brightness moments diverge. Hirata & Seljak (2003, H03 hereafter) address the divergence problems by realising an adaptive weighting scheme by minimising the error functional, sometimes dubbed the energy functional, (3)with the quadratic form (4)and the secondorder tensor (5)The tensor M is expressed in terms of the complex ellipticity e = e_{1} + ie_{2} and the size T of the image; f(ρ) is a weight function that HS03 chose to be a Gaussian weight f(ρ) = e^{− ρ/ 2}. In comparison to HS03, we have slightly changed the definition of ρ for convenience: here we use ρ instead of ρ^{2}. The set p = (A,x_{0},M), comprises a set of six parameters on which the functional E depends for a given galaxy image I.
Frequently another definition of complex ellipticity, the ϵellipticity, is more convenient (sometimes also known as the third flattening). It arises if we write ρ in the form (6)where V is symmetric. Obviously, we have V^{2} = M or . By writing V in the form (7)we see that V^{2} = M implies 2T = t^{2} (1 +  ϵ  ^{2}), and (8)We henceforth use ϵ as parametrisation of M because ϵ is an unbiased estimator of reduced shear in the absence of a PSF and pixellation (SS97). Conversely, the ellipticity e has to be calibrated with the distribution of unsheared ellipticities which poses another possible source of bias in a lensing analysis (H03).
As generally derived in Appendix A, the parameters p at the minimum of (3) are: (9)and (10)where is a constant. These equations are derived by HS03 for a Gaussian f(ρ), for which we have f′(ρ): = df/ dρ = − f/ 2. This shows that the bestfitting f(ρ) has the same centroid and, up to a scalar factor, the same second moment M = V^{2} as the with f′(ρ) adaptively weighted image I(x). This is basically also noted in Lewis (2009) where it is argued that the leastsquare fit of any sheared model to a preseeing image I(x) provides an unbiased estimate of the shear, even if it fits poorly.
In this system of equations, the centroid x_{0} and tensor M are implicitly defined because both f(ρ) and f′(ρ) on the righthandside are functions of the unknowns x_{0} and M: the weights adapt to the position, size, and shape of the image. Equations (9) and (10) therefore need to be solved iteratively. The iteration should be started at a point which is close to the final solution. Such a starting point could be obtained by using the image position, determined by the image detection software, as initial value for x_{0}, and tensor M determined from a circular weight function with the same functional form as f. Nevertheless, there is no guarantee that the solution of this set of equations is unique. In fact, for images with two brightness peaks one might suspect that there are multiple local minima in p of the functional E. This may occur, for instance, in the case of blended images. One standard solution to this particular problem is to identify blends and to remove these images from the shear catalogue. Alternatively we could in principle try to fit two template profiles to the observed image, that is by adding a second profile E_{2}(p_{2}  I) to the functional (3) and by minimising the new functional with respect to the parameter sets p and p_{2} of both profiles simultaneously.
2.2. Interpretation
If an image I(x) has confocal elliptical isophotes, with the same ellipticity for all isophotes, one can define the ellipticity of the image uniquely by the ellipticity of the isophotes. In this case, the ellipticity ϵ defined by the minimum of (3) coincides with the ellipticity of the isophotes for any weight f. We show this property in the following.
Assume that the brightness profile I(x) is constant on confocal ellipses so that we can write I(x) = S(ζ) where ζ: = (x − x_{c})^{T}B^{2}(x − x_{c}). Here x_{c} denotes the centre of the image, and the matrix elements of B describe the size and the shape of the image, in the same way as we discussed for the matrix V before. The function S(ζ) describes the radial brightness profile of the image. We start by writing (9) in the form (11)and introduce the transformed position vector z = B^{1}(x − x_{c}), or x = Bz + x_{c}. Then the previous equation becomes (12)where in terms of z the quadratic form ρ is (13)From these equations, we can see that x_{0} = x_{c} is the solution of (12) since then ρ is an even function of z, S is an even function of z, whereas the term in the parenthesis of (12) is odd, and the integral vanishes due to symmetry reasons. Thus we found that our adaptive moments approach yields the correct centre of the image.
Next, we rewrite (10) in the form (14)where β, see Eq. (10), is a constant factor (see Appendix A.3). We again used the transformation from x to z = B^{1}(x − x_{c}) and employed the fact that x_{0} = x_{c}. Accordingly, we have ρ = z^{T}BV^{2}Bz. We now show that the solution of (14) is given by V = λB with λ being a scalar factor. Using this Ansatz, we get ρ = λ^{2}  z  ^{2}, and (14) can be written, after multiplying from the left and from the right by B^{1}, as (15)Since both S and f′ in the numerator depend solely on  z  ^{2}, the tensor on the right hand side is proportional to the unit tensor 1, and (15) becomes a scalar equation for the scalar λ, (16)whose solution depends on the brightness profile S and the chosen weight function f. However, the fact that B differs from V only by the scalar factor λ implies that the derived ellipticity ϵ of V is the same as that of the elliptical image. Therefore we have shown that the approach of adapted moments recovers the true ellipticity with elliptical isophotes for any radial weight function f.
For a general brightness profile of the image, the interpretation of the GLAM ellipticity ϵ is less clear, and the ellipticity generally depends on the weight f. Nevertheless, ϵ is uniquely defined as long as a minimum of the functional (3) can be found. More importantly, for any weight f the GLAM ellipticity obeys the same simple transformation law under the action of gravitational shear, as shown in the following section.
2.3. Transformation under shear
We now consider the effect of a shear γ = γ_{1} + iγ_{2} and convergence κ on the GLAM ellipticity ϵ (Bartelmann & Schneider 2001). For an image with no noise, no pixellation, and no PSF convolution the ellipticity ϵ should be an unbiased estimate of the reduced shear g = g_{1} + ig_{2} = γ (1 − κ)^{1}. This is clearly true for sources that intrinsically have circular isophotes since the isophotes of the sheared images are confocal ellipses with an ellipticity ϵ = g. The minimum of (3) is hence at g by means of the preceding discussion.
For general images, let ϵ_{s} = ϵ_{s,1} + iϵ_{s,2} be the complex ellipticity of the image in the source plane and ϵ = ϵ_{1} + iϵ_{2} its complex ellipticity in the lens plane. We show now that for any brightness profile and template f(ρ), GLAM ellipticities have the extremely useful property to transform under the action of a reduced shear according to (17)This is exactly the wellknown transformation obtained from unweighted moments (SS97). To show this generally for GLAM , let I_{s}(y) be the surface brightness of an image in the source plane with source plane coordinates y. The centroid y_{0} and moment tensor M_{s} of I_{s} are defined by the minimum of (3) or, alternatively, by the analog of Eqs. (9) and (10) through (18)and (19)with (20)The ellipticity ϵ_{s} is given by (21)We now shear the image I_{s}. The shear and the convergence are assumed to be constant over the extent of the image, that is lens plane positions x are linearly mapped onto source plane positions y by virtue of where (22)x_{c} and y_{c} are such that the point x_{c} is mapped onto the point y_{c} by the lens equation, and both are chosen to be the central points around which the lens equation is linearised. We then find for the centroid y_{0} in the source plane (23)where . This then yields for (20) (24)In the next step, the expression (18) for the source centre can be rewritten by transforming to the image coordinates and using the conservation of surface brightness I_{s}(y(x)) = I(x): (25)With the same transformation, we rewrite the moment tensor (19) as (26)On the other hand, minimising the functional (3) for the surface brightness I(x) in the lens plane yields the expressions (9) and (10) for x_{0} and M, respectively. We then see that Eqs. (25), (26) and (9), (10) agree with each other, if we set (27)for which ρ_{s} = ρ. In particular, this shows that the centre x_{0} of the image is mapped onto the centre y_{0} of the source.
The relation (28)can be rewritten in terms of the square roots of the matrix M as (29)where is uniquely defined by requiring that for the symmetric, positivedefinite matrix M_{s}, V_{s} is symmetric and positivedefinite. Although both V and are symmetric, is in general not. Therefore V_{s} cannot be readily read off from (29). Instead, we use a rotation matrix (30)to write (31)If we now choose ϕ to be such that is symmetric, then . After a bit of algebra, we find that the rotation angle ϕ is given through (32)and we obtain as final result V_{s} as in (21) with (33)The inverse of (33) is given by (34)We recover for the GLAM ellipticity ϵ exactly the transformation law of unweighted moments (SS97). The GLAM ellipticity ϵ is therefore an unbiased estimator of the reduced shear g along the lineofsight of the galaxy, and there is no need to determine unweighted moments.
As a side remark, the transformation between ϵ and ϵ_{s} is a linear conformal mapping from the unit circle onto the unit circle, and from the origin of the ϵplane onto the point − g in the ϵ_{s}plane. If  g  > 1, then g has to be replaced by 1 /g^{∗} in Eq. (34), but we shall not be concerned here with this situation in the strong lensing regime.
2.4. Point spread function and pixellation
We have defined the GLAM ellipticity ϵ of an image I(x) relative to an adaptive weight f(ρ). This definition is idealised in the sense that it assumes an infinite angular resolution and the absence of any atmospheric or instrumental distortion of the image. Equally important, it ignores pixel noise. In this section, we move one step further to discuss the recovery of the original ϵ of an image after it has been convolved with a PSF and pixellated. The problem of properly dealing with noise in the image is discussed subsequently.
Let I_{pre}(x) be the original image prior to a PSF convolution and pixellation. This we call the “preseeing” image. Likewise, by the vector I_{post} of N_{pix} values we denote the “postseeing” image that has been subject to a convolution with a PSF and pixellation. For mathematical convenience, we further assume that I_{pre}(x) is binned on a fine auxiliary grid with N ≫ N_{pix} pixels of solid angle Ω. We list these pixel values as vector I_{pre}. The approximation of I_{pre}(x) by the vector I_{pre} becomes arbitrarily accurate for N → ∞. Therefore we express the postseeing image I_{post} = LI_{pre} by the linear transformation matrix L applied to the preseeing image I_{pre}. The matrix L with N × N_{pix} elements combines the effect of a (linear) PSF convolution and pixellation. Similarly, we bin the template f(ρ) in the preseeing frame to the grid of I_{pre}, and we denote the binned template by the vector f_{ρ}; as usual, the quadratic form ρ is here a function of the variables (x_{0},ϵ,t), Eq. (6). The GLAM parameters p_{pre} of the preseeing image are given by the minimum of E(p  I_{pre}), or approximately by (35)For the recovery of the preseeing ellipticity ϵ, the practical challenge is to derive the preseeing parameters p_{pre} from the observed image I_{post} in the postseeing frame. For this task, we assume that the transformation L is exactly known. We note that a linear mapping L is an approximation here; we ignore the nonlinear effects in the detector (Plazas et al. 2014; Gruen et al. 2015; Niemi et al. 2015; Melchior et al. 2015).
For a start, imagine a trivial case where no information is lost by going from I_{pre} to I_{post}. We express this case by a transformation L that can be inverted, which means that N = N_{pix} and L is regular. We then obtain p_{pre} by minimising E_{pre}(p  L^{1}I_{post}) with respect to p: we map I_{post} to the preseeing frame and analyse I_{pre} = L^{1}I_{post} there. This is equivalent to minimising the form (I_{post} − ALf_{ρ})^{T}(LL^{T})^{1}(I_{post} − ALf_{ρ}) in the postseeing frame.
Fig. 1
Examples of GLAM templates in the preseeing frame, f_{ρ} (top left), and the postseeing frame, Lf_{ρ} (other panels); the templates are Gaussian radial profiles with f(ρ) = e^{− ρ/ 2}. The bottom left panel simulates only pixellation, whereas the right column also shows the impact of a PSF, indicated in the top right corner, without (top) and with pixellation (bottom). 
For realistic problems where L^{1} does not exist, because N ≫ N_{pix}, this trivial case at least suggests to determine the minimum p_{post} of the new functional (36)as estimator of p_{pre}. This way we are setting up an estimator by forwardfitting the template f_{ρ} to the image in the postseeing frame with the matrix U being a metric for the goodness of the fit. Clearly, should L^{1} exist we recover (35) only by adopting U = (LL^{T})^{1}. So we could equivalently obtain p_{pre}, without bias, by fitting Lf_{ρ} to the observed image I_{post} in this case. However, realistically L is singular: the recovery of p_{pre} from (36) can only be done approximately. Then we could at least find an optimal metric to minimise the bias. We return to this point shortly. In any case, the metric has to be positivedefinite and symmetric such that always E_{post} ≥ 0. We note that the moments at the minimum of (36) are related but not identical to the adaptive moments in the postseeing frame. To obtain the latter we would fit a pixellated f_{ρ} with U = 1 to I_{post}. The bottom and top right images in Fig. 1 display examples of postseeing templates that are fitted to a postseeing image to estimate p_{pre} with the functional (36).
For singular L, the minimum of the functional yields an unbiased p_{pre} for any U if

1.
I_{pre}(x) has confocal elliptical isophotes with the radial brightness profile S(x);

2.
and if we choose f(ρ) = S(ρ) as GLAM template;

3.
and if E_{post} has only one minimum (nondegenerate).
To explain, due to 1. and 2. we find a vanishing residual (37)at the minimum of E_{pre} and consequently E_{pre}(p_{pre}  I_{pre}) = 0. At the same time for any metric U, we also have E_{post}(p_{pre}  I_{post}) = 0 because I_{post} − ALf_{ρ} = LR_{pre} = 0 for p = p_{pre}. Because of the lower bound E_{post} ≥ 0, these parameters p_{pre} have to coincide with a minimum of E_{post} and hence indeed p_{post} = p_{pre}. We note that the previous argument already holds for the weaker condition LR_{pre} = 0 so that a mismatch between I_{pre} and the template at p_{pre} produces no bias if the mapped residuals vanish in the postseeeing frame. In addition, if this is the only minimum of E_{post} then the estimator p_{post} is also uniquely defined (condition 3). An extreme example of a violation of condition 3 is the degenerate case N_{pix} = 1: the observed image consists of only one pixel. Then every parameter set (x_{0},ϵ,t) produces E_{post} = 0, if A is chosen correspondingly. But this should be a rare case because it is not expected to occur for N_{pix}> 6, thus for images that span over more pixels than GLAM parameters.
A realistic preseeing image is neither elliptical nor is our chosen template f(ρ) likely to perfectly fit the radial light profile of the image, even if it were elliptical. This mismatch produces a bias in p only if L is singular, and the magnitude of the bias scales with the residual of the template fit in the preseeing frame. To see this, let be the bestfitting template Af_{ρ} with parameters p_{pre}. The residual of the template fit in the preseeing frame is . In the vicinity of p_{pre}, we express the linear change of Af_{ρ} for small δp by its gradient at p_{pre}, (38)where (39)Each column G_{i} of the matrix G denotes the change of Af_{ρ} with respect to p_{i}. Therefore, close to p_{pre} we find the Taylor expansion (40)Furthermore, since p_{pre} is a local minimum of E_{pre}, Eq. (35), we find at the minimum the necessary condition (41)This means that the residual R_{res} is orthogonal to every G_{i}. Now let be the residual mapped to the postseeing frame. Then (36) in the vicinity to p_{pre} is approximately (42)which we obtain by plugging (40) into (36). This approximation is good if the bias is small, by which we mean that the minimum p_{post} of E_{post}(p  I_{post}) is close to p_{pre}. As shown in Aitken (1934) in the context of minimumvariance estimators, the firstorder bias δp_{min} = p_{post} − p_{pre} that minimises (42) is then given by (43)with U_{L}: = L^{T}UL. This reiterates that the bias δp_{min} always vanishes either for vanishing residuals R_{pre} = 0, or if L^{1} exists and we choose U = (LL^{T})^{1} as metric. The latter follows from Eq. (43) with U_{L} = 1 and Eq. (41). We note that δp_{min} does not change if we multiply U by a scalar λ ≠ 0. Thus any metric λU generates as much bias as U.
With regard to an optimal metric U, we conclude from Eq. (43) and G^{T}R_{pre} = 0 that we can minimise the bias by the choice of U for which U_{L} ≈ 1, or if ∥ L^{T}UL − 1 ∥ ^{2} is minimal with respect to the Frobenius norm ∥ Q ∥ ^{2} = tr^{(}QQ^{T}^{)}. This optimised choice of U corresponds to the socalled pseudoinverse (LL^{T})^{+} of LL^{T} (Press et al. 1992). The pseudoinverse is the normal inverse of LL^{T} if the latter is regular.
A practical computation of U = (LL^{T})^{+} could be attained by choosing a set of orthonormal basis function b_{i} in the preseeing frame, or approximately a finite number N_{base} of basis functions that sufficiently describes images in the preseeing frame. For every basis function, one then computes the images Lb_{i} and the matrix (44)The pseudoinverse of this matrix is the metric in the postseeing frame.
Specifically, for images that are only pixellated the matrix LL^{T} is diagonal. To see this, consider pixellations that map points e_{i} in the preseeing frame to a single pixels e′(e_{i}) in the postseeing frame, or Le_{i} = e′(e_{i}); both e_{i} and e′(e_{i}) are unit vectors from the standard bases in the two frames. According to (44), the matrix LL^{T} = ∑ _{i}e′(e_{i}) [ e′(e_{i}) ] ^{T} is then always diagonal, typically proportional to the unit matrix, and easily inverted to obtain the optimal metric U. Unfortunately, this U only makes the linearorder bias (43) vanish while we still can have a higherorder bias because of the singular L.
2.5. Similarity between modelbased and momentbased techniques
Through the formalism in the previous sections it becomes evident that there is no fundamental difference between modelbased techniques and those involving adaptive moments. Modelbased techniques perform fits of model profiles to the observed image, whereas momentbased ellipticities with the adaptive weight f′(ρ) are equivalently obtained from the image by fitting the ellipticial profile f(ρ) to the image. This requires, however, the existence of a unique minimum of the functional E(p  I) which we assume throughout the paper. We note that the similarity also extends to the special case of unweighted moments in Eqs. (1) and (2) which are in principle obtained by fitting f(ρ) = ρ since f′(ρ) = 1 in this case.
Nonetheless one crucial difference to a modelfitting technique is that a fit of f(ρ) does not assume a perfect match to I(x): the functional E(p  I) needs to have a minimum, but the fit is allowed to have residuals R_{pre}, this means E(p  I) ≠ 0 at the minimum. As shown, for the estimator based on (36) this may cause bias, Eq. (43), but only when analysing postseeing images hence for L ≠ 1. In practice, different methodologies to estimate the preseeing momentbased ellipticity certainly use different approaches. Our choice of a forwardfitting estimator (36) is very specific but is optimal in the sense that it is always unbiased for LR_{pre} = 0 or regular L. Yet other options are conceivable. For instance, we could estimate moments in the postseeing frame first and try to map those to the preseeing frame (e.g. H03 or Melchior et al. 2011, which aim at unweighted moments). It is then unclear which weight is effectively applied in the preseeing frame. Therefore the expression Eq. (43) for the bias is strictly applicable only to our estimator and adaptive moments. But it seems plausible that the bias of estimators of preseeing moments generally depends on the residual R_{pre} since the brightness moments x_{0,i} and M_{ij} are the solutions to a bestfit of elliptical templates.
In the literature the problem of bias due to residuals in model fits is known as model bias or underfitting bias (Zuntz et al. 2013; Bernstein 2010). Consequently, momentbased techniques are as prone to underfitting bias as modelbased methodologies.
3. Statistical inference of ellipticity
Realistic galaxy images I are superimposed by instrumental noise δI. Therefore the preseeing GLAM ellipticity can only be inferred statistically with uncertainties, and it is, according to the foregoing discussion, subject to underfitting bias. For a statistical model of the ellipticity ϵ, we exploit the previous conclusions according to which the ellipticity of I(x) for the adaptive weight f′(ρ) is equivalent to ϵ of the bestfitting template f(ρ). This renders the inference of ϵ a standard forwardfit of a model ALf(ρ) to I.
We consider postseeing images I with Gaussian noise δI, that means images I = I_{post} + δI. The covariance of the noise is N = ⟨ δIδI^{T} ⟩, while I_{post} = LI_{pre} is the noisefree image in the postseeing frame. A Gaussian noise model is a fair assumption for faint galaxies in the skylimited regime (Miller et al. 2007). Possible sources of noise are: readout noise, sky noise, photon noise, or faint objects that blend with the galaxy image. If an approximate Gaussian model is not applicable, the following model of the likelihood has to be modified accordingly.
The statistical model of noise are given by the likelihood ℒ(I  p) of an image I = LI_{pre} + δI given the GLAM parameters p. We aim at a Bayesian analysis for which we additionally quantify our prior knowledge on parameters by the PDF P_{p}(p). We combine likelihood and prior to produce the marginal posterior (45)of ellipticity by integrating out the nuisance parameters (x_{0},A,t); the constant normalisation of the posterior is irrelevant for this paper but we assume that the posterior is proper (it can be normalised). Our choice for the numerical experiments in this study is a uniform prior P_{p}(p) for positive sizes t and amplitudes A, ellipticities  ϵ  < 1, and centroid positions x_{0} inside the thumbnail image. As known from previous Bayesian approaches to shear analyses, the choice of the prior affects the consistency of the ellipticity posteriors (see, e.g. BA14). The origin of the priordependence will become clear in Sect. 3.3.
With regard to notation, we occasionally have to draw random numbers or vectors of random numbers x from a PDF P(x) or a conditional density P(x  y). We denote this by the shorthand x ~ P(x) and x ~ P(x  y), respectively. As common in statistical notation, distinct conditional probability functions may use the same symbol, as for instance the symbol P in P(x  y) and P(y  x).
3.1. Caveat of point estimates
Fig. 2
Toymodel demonstration of a maximumlikelihood estimator (red), a maximumlikelihood estimator with firstorder bias correction (green), and an estimator exploiting the full posterior (blue). Data points display the estimator average (yaxis) over 10^{6} data points at varying signaltonoise levels (xaxis). The true value to be estimated is x = 1. The panels show different signaltonoise regimes; − nppt denotes y = 1 − n/ 10^{3}. 
The bias in a lensing analysis is not only affected by how we statistically infer galaxy shapes but also how we process the statistical information later on. To demonstrate in this context the disadvantage of point estimators in comparison to a fully Bayesian treatment, we consider here a simplistic nonlinear toy model. This model has one parameter x and one single observable y = x^{3} + n that is subject to noise n. By n ~ N(0,σ) we draw random noise from a Gaussian distribution N(0,σ) with mean zero and variance σ. From the data y, we statistically infer the original value of x. Towards this goal we consider the (log)likelihood of y given x which is − 2lnℒ(y  x) = (y − x^{3})^{2}σ^{2} + const.
A maximum likelihood estimator of x is given by x_{est} = y^{1 / 3}, the maximum of ℒ(y  x). We determine the bias of x_{est} as function of signaltonoise ratio (S/N) x/σ by averaging the estimates of N_{real} = 10^{6} independent realisations of y. The averages and the standard errors are plotted as red line in Fig. 2. Clearly, x_{est} is increasingly biased low towards lower S/N levels. In the context of lensing, this would be noise bias. As an improvement we then correct the bias by employing the firstorder correction in Refregier et al. (2012) for each realisation of y. As seen in the figure, this correction indeed reduces the systematic error, but nevertheless breaks down for S/N ≲ 3.
On the other hand in a fully Bayesian analysis, we obtain constraints on x that are consistent with the true value for any S/N. For this purpose, we make N_{real} independent identically distributed realisations (i.i.d.) y_{i} and combine their posterior densities P_{post}(x  y_{i}) ∝ ℒ(y_{i}  x) P_{prior}(x) by multiplying the likelihoods; we adopt a uniform (improper) prior P_{prior}(x) = const. This gives us for y = (y_{1},...,y_{Nreal}), up to a normalisation constant, the combined posterior (46)As expected due to the asymptotic normality of posteriors (under regularity conditions), for i.i.d. experiments y_{i} the product density is well approximated by a Gaussian N(x_{0},σ_{x}) and is consistent with the true value x (van der Vaart 1998). We plot values of x_{0} and σ_{x} in Fig. 2 as blue data points.
In conclusion, keeping the full statistical information P_{post}(x  y) in the inference of x yields consistent constraints over the entire S/N range probed: the noise bias vanishes. Also note that the Bayesian approach has not substantially increased the error σ_{x} compared to the error of the point estimator (relative sizes of error bars); both approaches have similar efficiency.
3.2. Likelihood model and underfitting bias
Inspired by the foregoing Bayesian toy model that is free of noise bias, we set up a Bayesian approach for GLAM ellipticities. To construct a likelihood for a GLAM fit in the preseeing frame, we first consider, similar to Sect. 2.4, the trivial case where L is regular. This is straightforward since we can map the noisy image I → L^{1}I back to the preseeing frame and determine the noise residual for given preseeing p, (47)The inverse noise covariance in the preseeing frame is L^{T}N^{1}L. The logarithmic likelihood of δI_{pre}(p) in the Gaussian case is thus (48)where “const.” expresses the normalisation of the likelihood. Therefore, we can equivalently write the preseeing fit in terms of available postseeing quantities. We note here that the transform LR_{pre} of the (unknown) preseeing residual is for singular L and R_{pre} ≠ 0 not equal to the postseeing residual R_{post}, which we defined in a leastsquare fit of ALf_{ρ} to I_{post} (see Sect. 2.4).
In reality, L is singular so the previous steps cannot be applied. Nevertheless, Eq. (48) is the correctly specified likelihood of a forward fit if the model m(p): = Af_{ρ} + R_{pre} perfectly describes the brightness profile I_{pre} for some parameters p_{true}. Therefore we expect no inconsistencies for singular L as long as the correct R_{pre} can be given. Since R_{pre} is unknown, however, we investigate in the following the impact of a misspecified likelihood that does not properly account for our ignorance in the preseeing residuals. We do this by assuming R_{pre} ≡ 0 and employing (49)Obviously, this is a reasonable approximation of (48) if ∥ LR_{pre} ∥ ≪ ∥ I ∥ so that residuals are only relevant if they are not small when compared to the image in the postseeing frame. It also follows from Eq. (48) that for LR_{pre} = 0 the likelihood is correctly specified even if R_{pre} ≠ 0 and L being singular, similar to the noisefree case. More generally we discuss later in Sect. 5 how we could modify the likelihood ℒ(I  p) to factor in our insufficient knowledge about R_{pre}. Until then the approximation (49) introduces underfitting bias into the likelihood model, making it inconsistent with the true p_{pre}.
To show the inconsistency, we proceed as in the foregoing section on the toy model. We consider a series of n i.i.d. realisations I_{i} of the same image I_{post} and combine their likelihoods ℒ(I_{i}  p) for a given p into the joint (product) likelihood (50)and we work out its limit for n → ∞. The joint likelihood can be written as (51)Here we have made use of the properties of the trace of matrices, namely its linearity and that . For large n, we can employ the asymptotic expressions (52)to the following effect: (53)Thus the joint likelihood peaks for n → ∞ at the minimum p_{post} of ∥ I_{post} − ALf_{ρ} ∥ _{N}. This is equivalent to the location of the minimum of E_{post}(p  I_{post}), Eq. (36), with metric U = N^{1}. Consequently, as in the noisefree case, we find an inconsistent likelihood if the residual R_{pre} is nonvanishing and if L^{T}N^{1}L is not proportional to the unity matrix. We additionally expect a smaller bias δp = p_{post} − p_{pre} for smaller levels of residuals^{1}.
With regards to the noise dependence of underfitting bias, we find that δp does not change if we increase the overall level of noise in the image I. This can be seen by scaling the noise covariance N → λ N with a scalar λ. Any value λ> 0 results in the same minimum location for E_{post}(p  I_{post}) so that δp is independent of λ: there is no noise bias.
Moreover, the bias δp of a misspecified likelihood seems to depend on our specific assumption of a Gaussian model for the likelihood. It can be argued on the basis of general theorems on consistency and asymptotic normality of posteriors, however, that for n → ∞ we obtain the same results for other likelihood models under certain regularity conditions (van der Vaart 1998; Appendix B in Gelman et al. 2013). The latter requires continuous likelihoods that are identifiable, hence ℒ(I  p_{1}) ≠ ℒ(I  p_{2}) for p_{1} ≠ p_{2}, and that the true p_{true} is not at the boundary of the domain of all parameters p. This is stricter than our previous assumptions for the Gaussian model where we needed only a unique global maximum of the likelihood ℒ(I_{post}  p). It could therefore be that a unique maximum is not sufficient for nonGaussian likelihoods.
3.3. Prior bias
The foregoing section discusses the consistency of the likelihood of a single image I. We can interpret the analysis also in a different way: if we actually had n independent exposures I_{i} of the same preseeing image, then combining the information in all exposures results in a posterior that is consistent with p_{post} for a uniform prior P_{p}(p) = 1. As discussed in van der Vaart (1998), the more general Bernsteinvon Mises theorem additionally shows that under regularity conditions the choice of the prior is even irrelevant provided it sets prior mass around p_{post} (Cromwell’s rule). This might suggest that for a correctly specified likelihood ℒ(I_{i}  p), a fully Bayesian approach for the consistent measurement of ϵ might be found that is independent of the specifics of the prior and has no noise bias in the sense of Sect. 3.2. This is wrong as shown in the following.
Namely, in contrast to the previous simplistic scenario, sources in a lensing survey have varying values of p: they are intrinsically different. For more realism, we therefore assume now i = 1...n preseeing images that, on the one hand, shall have different centroid positions x_{0,i}, sizes t_{i}, amplitudes A_{i} but, on the other hand, have identical ellipticities ϵ. Our goal in this experiment is to infer ϵ from independent image realisations I_{i} = I_{post,i} + δI_{i} by marginalising over the 4n nuisance parameters q_{i} = (x_{0,i},t_{i},A_{i}). This experiment is similar to the standard test for shear measurements where a set of different preseeing images is considered whose realisations I_{i} are subject to the same amount of shear (e.g. Bridle et al. 2010). As a matter of fact, the inference of constant shear from an ensemble of images would just result in 2n additional nuisance parameters for the intrinsic shapes with essentially the same following calculations.
Let P_{q}(q_{i}) be the prior for the four nuisance parameters q_{i} of the ith image and P_{ϵ}(ϵ) = 1 a uniform prior for ϵ. We combine the GLAM parameters in p_{i}: = (q_{i},ϵ), and we assume that all images have the same prior density and that the noise covariance N applies to all images. The marginal posterior of ϵ is then the integral (54)with being a normalization constant.
The product of the likelihood densities inside the integral is given by (55)Here we have taken into account that the GLAM parameters partly differ, indicated by the additional index in A_{i} and f_{ρ,i}. This is different in Eq. (53) where we take the product of full likelihoods in pspace without marginalization. In the limit of n → ∞, we find in addition to the relations (52) that (56)because δI_{i} is uncorrelated to ALf_{ρ,i} so that vanishes on average for many δI_{i}. Therefore, for the asymptotic statistic we can replace all I_{i} by I_{post,i} in Eq. (55) to obtain (57)and, as a result, for Eq. (54) (58)where P_{ϵ}(ϵ  I_{post,i}) is the marginal ellipticity posterior for the ith noisefree image.
The limit (58) has the interesting consequence that the consistency with p_{post} of the marginal posterior depends on the specific choice of the prior density P_{q}(q_{i}). To show this, consider one particular case in which, for simplicity, all preseeing images are identical such that I_{post,i} ≡ I_{post}. Then, according to (58), the ellipticity posterior converges in distribution to [ P_{ϵ}(ϵ  I_{post}) ] ^{n} which for n → ∞ peaks at the global maximum of P_{ϵ}(ϵ  I_{post}). It is then easy to see that we can always change the position of this maximum by varying the prior density in P_{ϵ}(ϵ  I_{post}). In particular, even if the likelihoods are correctly specified, we generally find an inconsistent marginal posterior depending on the prior. A similar argument can be made if I_{post,i} ≠ I_{post,j} for i ≠ j.
Fig. 3
Prior bias in the marginal posterior P_{ϵ}(ϵ  I_{1},...,I_{n}) as function of S/N ν for different galaxy sizes r_{h} (in arcsec). The posterior assumes a uniform prior. Shown is the error δϵ =  ϵ − ϵ_{true}  of the inferred ϵ for a true ϵ_{true} = 0.3 as obtained by combining the marginal posteriors of 5 × 10^{3} exposures of the same galaxy with random centroid positions. The pixel size 0.1 arcsec equals the PSF size (Moffat). Galaxy profiles and GLAM templates have a Sérsic profile with n = 2: there is no underfitting. 
For a concrete example, we perform for Fig. 3 a simulated analysis of 5 × 10^{3} noisy images with ϵ_{true} = 0.3. All preseeing images are identical to one particular template Af_{ρ} (Sérsic profile with n = 2). Therefore we have a correctly specified likelihood model and no underfitting. We adopt a uniform prior P_{q}(q) (and P_{ϵ}(ϵ)). The details on the simulated images and their analysis are given in Sect. 4. For each data point, we plot the mean and variance of the marginal posterior P_{ϵ}(ϵ  I_{1},...,I_{n}) relative to the true ellipticity ϵ_{true} as function of S/N ν and for different image sizes r_{h}. Evidently, the bias δϵ increases for smaller ν thereby producing a noisedependent bias which is not present when analysing the consistency of ℒ(I  p) of individual images as in Sect. 3.2.
The sensitivity of the posterior to the prior implies that consistency (for a correctly specified likelihood) could be regained by choosing an appropriate prior density P_{q}(q_{i}). On the other hand, the dependence on the prior runs against the conventional wisdom that the prior should become asymptotically irrelevant for n → ∞ as the joint likelihood starts dominating the information on inferred parameters. Indeed, general theorems show this under certain regularity conditions (see e.g. discussion in Chap. 4 of Gelman et al. 2013). These conditions are, however, not given here because the total number of model parameters is not fixed but rather increases linearly with n due to a new set of nuisance parameters q_{i} for every new galaxy image I_{i}. The observed breakdown of consistency is the result. A trivial (but practically useless) prior to fix the problem is one that puts all prior mass at the true values of the nuisance parameters which essentially leaves only ϵ as free model parameter. Likewise, an analysis with sources that knowingly have the same values for the nuisance parameters also yields consistent constraints; this is exactly what is done in Sect. 3.2. A nontrivial solution, as reported by BA14, is to use “correct priors” for the nuisance parameters that are equal to the actual distribution of q in the sample. On the downside, this raises the practical problem of obtaining correct priors from observational data.
In summary, for an incorrect prior of q, such as our uniform prior, we find a noisedependent bias in the inferred ellipticity despite a fully Bayesian approach and a correctly specified likelihood. We henceforth call this bias “prior bias”. This emphasises the difference to the noise bias found in the context of point estimators.
4. Simulated shear bias
We have classified two possible sources of bias in our Bayesian implementation of adaptive moments: underfitting bias and prior bias. In this section, we study their impact on the inference of reduced shear g by using samples of mock images with varying S/N, galaxy sizes, and galaxy brightness profiles. In contrast to our previous experiments, the images in each sample have random intrinsic shapes but are all subject to the same amount of shear, and we perform numerical experiments to quantify the bias. Moreover, this section outlines practical details of a sampling technique for the posteriors of GLAM ellipticities and the reducedshear, which is of interest for future applications (Sects. 4.4 and 4.5).
The PSF shall be exactly known for these experiments. For a large shear bias, we choose a relatively small PSF size and galaxy images that are not much larger than a few pixels. According to the discussion in Sect. 2.4, underfitting bias is mainly a result of pixellation which mathematically cannot be inverted. We note that a larger PSF size would reduce the underfitting bias since it spreads out images over more image pixels. Overall the values for shear bias presented here are larger than what is typically found in realistic surveys (e.g. Zuntz et al. 2013). We start this section with a summary of our simulation specifications.
4.1. Pointspread function
If not stated otherwise, our simulated postage stamps consist of 20 × 20 pixels (squares), of which one pixel has a size of 0.1′′; the simulated galaxies are relatively small with a halflight radius r_{h} of only a few times the pixel size (0.15′′ ≤ r_{h} ≤ 0.3′′). Additionally, we adopt an isotropic Moffat PSF, (59)with the full width half maximum (FWHM) of θ_{FWHM} = 0.1′′ and β = 5 (Moffat 1969).
4.2. GLAM template profiles
As GLAM templates we employ truncated Sérsiclike profiles with index n, (60)and (61)(Capaccioli 1989). To avoid a numerical bias due to aliasing at the edges of the grid during the Fourier transformation steps in the sampling code, we have introduced the auxiliary function h(x) that smoothly cuts off the Sérsic profile beyond ρ ≈ 9. If f(ρ) were the radial light profile of an elliptical galaxy, the truncation would be located at about three halflight radii.
We use template profiles with n = 2 throughout. This index n falls between the values of n used for the model galaxies and thus is a good compromise to minimise the fit residual and thereby the underfitting bias.
4.3. Mock galaxy images
We generate postage stamps of mock galaxy images with varying halflight radii r_{h}, radial light profiles, and signaltonoise ratios ν. To this end, we utilise the code that computes the postseeing GLAM templates (Appendix B). We consider preseeing images of galaxies with elliptical isophotes of three kinds of light profiles, Eq. (60): (1) exponential profiles with Sérsic index n = 1 (EXP; exponential), (2) deVaucouleur profiles with n = 4 (DEV; bulgelike) and (3) galaxies with profiles n = 2 that match the profile of the GLAM template (TMP; templatelike). TMP galaxies hence cannot produce underfitting bias.
We devise uncorrelated Gaussian noise in the simulation of postage stamps. To determine the RMS variance σ_{rms} of the pixel noise for a given ν, let f_{i} be the flux inside image pixels that is free of noise and f_{tot} = ∑ _{i}f_{i} the total flux. From this, we compute a halflight flux threshold f_{th} defined such that the integrated flux f_{hl} = ∑ _{fi ≥ fth}f_{i} above the threshold is f_{hl} = f_{tot}/ 2 or as close as possible to this value. The pixels i with f_{i} ≥ f_{th} are defined to be within the halflight radius of the image; their number is N_{hl}: = ∑ _{fi ≥ fth}; the integrated noise variance within the halflight radius is . The signaltonoise ratio within the halflight radius is therefore , or (62)Figure 4 depicts four examples with added noise for different ν.
Fig. 4
Examples of simulated images of galaxies with random ellipticities. Signaltonoise ratios from left to right and top to bottom: ν = 10, 20, 40, and 200. The radial light profile of the preseeing images is exponential with r_{h} = 0.2′′; the pixel size is 0.1′′. The FWHM of the PSF is indicated by the circle in the top right corners. 
To simulate galaxy images with intrinsic ellipticities that are sheared by g, we make a random realisation of an intrinsic ellipticity ϵ_{s} and compute the preseeing ellipticity with (17). As PDF for the intrinsic ellipticities we assume a bivariate Gaussian with a variance of σ_{ϵ} = 0.3 for each ellipticity component; we truncate the PDF beyond  ϵ_{s}  ≥ 1.
In realistic applications, centroid positions of galaxy images are random within an image pixel, so we average all our following results for bias over subpixel offsets within the quadratic solid angle of one pixel at the centre of the image. This means: in an sample of mock images, every image has a different subpixel position which is chosen randomly from a uniform distribution. In this averaging process, care has to be taken to perform an even sampling of subpixel offsets. To ensure this with a finite number of random points in all our following tests, we employ a subrandom Sobol sequence in two dimensions for the random centroid offsets (Press et al. 1992).
4.4. MonteCarlo sampling of ellipticity posterior
For practical applications of a Bayesian GLAM analysis, we produce a Monte Carlo sample of the posterior P_{ϵ}(ϵ  I), Eq. (45) with the approximate likelihood ℒ(I  p) in Eq. (49). This sample consists of a set (ϵ_{i},w_{i}) of 1 ≤ i ≤ N_{real} pairs of values ϵ_{i} and w_{i}, which determine the sampling position ϵ_{i} and a sampling weight w_{i}. For this paper, we use N_{real} = 50 sampling points. In contrast to a lensing analysis with singlevalued point estimators of ellipticity, a Bayesian analysis employs the sample (ϵ_{i},w_{i}) of each galaxy. To attain the sample (ϵ_{i},w_{i}) we invoke the importance sampling technique (e.g. Marshall 1956; Kilbinger et al. 2010).
For this technique, we define an approximation Q(p) of P_{p}(p  I), the socalled importance distribution function, from which we draw a random sample p_{i}. The ellipticity component ϵ_{i} of p_{i} is then assigned the weight w_{i} = P_{p}(p_{i}  I) Q(p_{i})^{1} which we normalise to ∑ _{i}w_{i} = 1 afterwards. As importance function, we use a multivariate Gaussian with mean at the maximum p_{ml} of the likelihood ℒ(I  p) and a covariance defined by the inverse Fisher matrix F^{1} at the maximum (Fisher 1935; Tegmark et al. 1997). More implementation details are given in Appendix B.
If Q(p) is too different from P_{p}(p  I), the sample will be dominated by few points with large weights, usually due to sampling points in the tail of the posterior for which the importance function predicts a too low probability, this means w_{i} becomes large. This is indicated by a small effective number of points compared to N_{real}. This can produce a poor convergence and thus extreme outliers in the analysis that merely appear to have tight constrains on ϵ. Typically affected by this are images that are both small compared to the pixel size and low in signaltonoise. These images tend to exhibit posteriors that allow parameter solutions with small sizes t compared to the true t or high eccentricities  ϵ  ≳ 0.7, giving the posterior a complex, distinctly nonGaussian shape. In future applications, this problem can be addressed by finding a better model of the importance function.
For the scope of this paper, we address this problem at the expense of computation time by increasing the number of sampling points and by resampling. This means: for MonteCarlo samples with N_{eff}<N_{real}/ 2, we draw more samples from the importance distribution until N_{eff} ≥ N_{real}/ 2 in the expanded sample. By an additional rule, we stop this process if the expanded sample size becomes too large and reaches 10 × N_{real}. This case is indicative of a failure of the importance sampling. If this happens, we switch to a timeconsuming standard Metropolis algorithm to sample the posterior with 10^{3} points after 10^{2} preliminary burnin steps (Metropolis et al. 1953). This technique does not assume a particular shape for the posterior but performs a random walk through the p space, thereby producing a correlated sample along a MonteCarlo Markov chain. As symmetric proposal distribution for this algorithm, we adopt a multivariate Gaussian with covariance 1.5 × F^{1}; all points of the chain have equal weight w_{i}; the starting point of the chain is p_{ml}. Finally, in the resampling phase, we bootstrap the expanded or Metropolis sample by randomly drawing N_{real} points from it with probability w_{i} (with replacement). All selected data points are given the equal weights in the final catalogue.
4.5. Posterior density of reduced shear
For the inference of g, we convert the ellipticity posterior P_{ϵ}(ϵ  I) into a posterior P_{g}(g  I) of shear. To this end, let us first determine the P_{g}(g  ϵ) for an exactly known ϵ. We express our uncertainty on the intrinsic ellipticity ϵ_{s} by the prior P_{s}(ϵ_{s}), and P_{g}(g) is our apriori information on g. The values of g and ϵ_{s} shall be statistically independent, by which we mean that P_{sg}(ϵ_{s},g) = P_{s}(ϵ_{s}) P_{g}(g)^{2}. Applying a marginalization over ϵ_{s} and then Bayes’ rule, we find (63)or, equivalently, By we denote the normalisation of P_{g}(g  ϵ), (66)The normalisation only depends on the modulus of ϵ for isotropic priors for symmetry reasons. The integration over the Dirac delta function δ_{D}(x) uses Eq. (17). The determinant  d^{2}ϵ_{s}/ d^{2}ϵ , the Jacobian of the linear conformal mapping ϵ_{s}(g,ϵ), is a function of ϵ and g. For the weak lensing regime  g  ≤ 1 of interest, this is (67)(Geiger & Schneider 1998). To now account for the measurement error of ϵ in the shear posterior, we marginalise P_{g}(g  ϵ), Eq. (65), over ϵ with P_{ϵ}(ϵ  I) as error distribution, (68)In our Monte Carlo scheme, we sample the ellipticity posterior of every galaxy i through (ϵ_{ij},w_{ij}) ~ P_{ϵ}(ϵ  I_{i}) by 1 ≤ j ≤ N_{real} values ϵ_{ij} of ellipticity and statistical weight w_{ij}. To convert this sample to an approximation of the g posterior, we replace P_{ϵ}(ϵ  I_{i}) inside the integral (68) by the point distribution Σ_{j}w_{ij}δ_{D}(ϵ − ϵ_{ij}), (69)For the following, we adopt a uniform prior for g, which means P_{g}(g) ∝ H(1 −  g ). For reasons discussed in Sect. 3.3, the prior on the nuisance ϵ_{s} presumably has to be equal to the distribution of intrinsic shapes for a consistent shear posterior. To investigate the sensitivity of shear bias as to P_{s}(ϵ_{s}) we therefore use two types of priors: the correct prior, which is the true intrinsicshape distribution in Sect. 4.3, or a uniform prior H(1 −  ϵ_{s} ). For each prior P_{s}(ϵ_{s}), we numerically compute for different ϵ once and interpolate between them later on in (69). We finally combine the posteriors P_{g}(g  I_{i}) of all images I_{i} in the sample by multiplying posterior values on a ggrid. For this, we apply (69) to every galaxy in the sample independently.
4.6. Results
Fig. 5
Plots of the multiplicative bias m for simulated images, based on Eq. (69), for two different priors P_{s}(ϵ_{s}) (filled data points: correct prior; open data points: uniform prior). Different styles for the data points indicate different image sizes r_{h}, see key inside figure, while colours vary with galaxy type: GLAM template (TMP; red); exponential (EXP; green); bulgelike (DEV; blue). Data points for r_{h} = 0.3′′, same galaxy type, and same prior are connected by dotted lines to guide the eye. The prior for galaxy sizes t, amplitudes A, and centroids x_{0} is uniform giving rise to noisedependent prior bias; the constant offset of EXP and DEV is due to underfitting. A square image pixel has the size 0.1′′, which also equals the PSF size. Results for m for a larger PSF size can be found in Table 1. 
For each experiment, we consider a sample of statistically independent galaxy images that are subject to the same reduced shear g but have random intrinsic ellipticities ϵ_{s}. To infer g from in a fully Bayesian manner, we compute the posteriors P_{g}(g  I_{i}) of g for every image I_{i} separately with Eq. (69) and then combine all posteriors by the product (70)This test is related, but not identical, to the ring test in Nakajima & Bernstein (2007): we do not fix the ellipticity magnitude  ϵ_{s}  during one test run but instead use a random sample of intrinsic ellipticities. Each galaxy sample consists of n = 5 × 10^{4} simulated noisy galaxy images to typically attain a precision for g of the order 10^{3}. This number applies for the nonuniform prior P_{s}(ϵ_{s}). The precision is lower for the less informative uniform prior. Moreover, for each galaxy with intrinsic ϵ_{s} we also use − ϵ_{s} in another image belonging to the sample. By doing so the mean of intrinsic ellipticities in an sample is always zero, which substantially reduces the numerical noise in the result. For every set of image parameters, we vary the input shear between the three values g_{true} ∈ { 0, ± 0.05, ± 0.1 }. For these five measurements, let ⟨ g  g_{true} ⟩ be the mean of the posterior (70) for a given g_{true} of the sample. We then determine the multiplicative bias m and the additive bias c in (71)by bestfitting m and c to the mean and dispersion of (70) for the three input values of g_{true}; m shall be a realvalued number.
In Fig. 5, we plot m (filled data points) juxtaposed with the corresponding values based on a uniform prior (open data points); same data points indicate the same size r_{h} while colours, varying by row, indicate the light profiles of the images. For the numbers in the table and the data points in the figure, error bars quantify a 68% posterior interval of the measurement around the quoted mean. The size of error bars account for pixel noise and shape noise. For the range of ν studied here, the impact of pixel noise is subdominant: the errors increase only between 10% and 20% from ν = 200 to ν = 10; the increase is larger for small images. Values for the additive bias c (not shown) for all profiles are small within a few 10^{4}, except at ν = 10 for DEV with r_{h} = 0.15′′,0.2′′ and for TMP at 0.15′′: there c can reach negative amplitudes of the order of 10^{3}. This indicates that an additive bias could be relevant for image sizes close to the pixel size.
Multiplicative bias m in per cent for three profiles TMP, EXP, DEV, and sizes r_{h} = 0.2′′ only.
In contrast, a multiplicative bias m is typically significant. At high S/N ν = 200, this bias in Fig. 5 is mainly underfitting bias which is consequently absent for TMP galaxies since their likelihood is correctly specified. The shear of bulgelike galaxies DEV, with steeper slope compared to f(ρ), is underestimated by m = − 9.6 ± 0.5%; the shear of exponential galaxies EXP, which have a shallower slope, is overestimated by m = + 7.7 ± 0.5%. These numbers are the mean and standard error for all galaxy sizes at ν = 200; they depend on the specific PSF and pixel sizes chosen for this experiment.
Compared to the underfitting bias at ν = 200, the bias m significantly drops if ν ≲ 20 for all galaxies, most prominently at r_{h} = 0.15′′, but stays roughly constant within a couple of per cent otherwise. We attribute the noisedependence of m to prior bias owing to our choice of an (incorrect) uniform prior for the nuisance parameters q of the images (see Sect. 3.3). The prior bias thus becomes only relevant here at sufficiently low S/N. The particular value of S/N is specific to the PSF size. This can be seen in Table 1 which lists estimates for m for galaxies, in a slightly smaller sample n = 2 × 10^{4}, with size r_{h} = 0.2′′ but for a modified, roughly twice as large PSF size 0.22′′. Now the noisedependence of prior bias is already visible below ν ≲ 40. On the other hand, the underfitting bias, to be read off at high S/N ν = 200, is roughly halved since the images are spread out over more pixels.
The choice of a uniform prior for the intrinsic ellipticities, distinctively different from the true distribution of ϵ_{s} in the sample, has a negligible effect on shear bias here for both PSF sizes. Values of m between the two priors P_{s}(ϵ_{s}) are consistent, as found by comparing filled and open data points of same colour and at same ν in Fig. 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 apriori uncertainty on ϵ_{s}. This implies that a shear bias due to an incorrect prior P_{s}(ϵ_{s}) 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 f′(ρ): they (i) are identical with the ellipticity of isophotes of elliptical galaxy images; (ii) under the influence of shear they behave exactly like ϵ 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 ϵ 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 (x_{0},α M_{I}) of the bestfitting template f(ρ) will, for a constant scalar α, be the adaptively weighted moments (x_{0},M_{I}) of the galaxy light profile I(x) for the weight f′(ρ) (Appendix A). This means that the radial profile of the weight at separation r from the centroid is w(r) ∝ r^{1}df_{r}(r) / dr for a template profile f_{r}(r): = f(r^{2}). We exploited this relation between adaptive moments and moments of a bestfitting elliptical profile to analyse limits to adaptive momentbased 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 postseeing image. It potentially arises because of a loss of information on the preseeing galaxy image, fundamentally due to pixellation. To explore the limitations, we assume noisefree images and express in Sect. 2.4 the mapping between a preseeing and postseeing image by the linear operation L. The preseeing adaptive moments are equivalent to a leastsquare fit with residual R_{pre} of an elliptical profile f(ρ) to the preseeing image I(x). If estimated from the postseeing image whilst ignoring residuals, the inferred ellipticity is biased for LR_{pre} ≠ 0and a singular L. The latter indicates a loss of information on the preseeing image. For noisefree images and the estimator (36) the bias is, to linear order, given by Eq. (43). In (36) the bias is optimally reduced through the metric U = (LL^{T})^{+}. With this metric the estimator is unbiased for regular L, hence in principle also for an invertible convolution with an anisotropic PSF; invertible convolutions have kernels K(x) with in the Fourier domain. Practically, however, a singular mapping is always present due to pixellation so that the quadratic estimator (36) could never fully remove underfitting bias.
The inference of GLAM ellipticity from noisy images with a likelihood that ignores fitting residuals produces underfitting bias; the underfitting bias depends on the correlation of pixel noise and its homogeneity over the image. To deal with pixel noise in images, we have put GLAM in a Bayesian setting in Sect. 3 by employing an approximate model (49) for the likelihood that ignores fit residuals and thus equals an imperfect model fit with an elliptical profile. We use this to explore the impact of a misspecified likelihood that, as a result, can be shown to be subject to underfitting bias. The bias is revealed in the limit of combining the likelihoods of n → ∞ independent exposures of the same image (see Sect. 3.2). To lowest order in R_{pre}, we find the bias to vanish if L^{T}N^{1}L is proportional to the unit matrix. The underfitting bias is unchanged for a rescaled noise covariance: there hence is no noise bias from a misspecified likelihood. Additionally, the magnitude of bias depends on the details of N and can also emerge for L = 1 if N is not proportional to the unit matrix, which is the case for heterogeneous or correlated noise. Due to the close relation of GLAM ellipticity to other definitions of ellipticity we expect similar dependencies of bias on the noise covariance there, which may impact future calibration strategies in lensing applications because they apparently should not ignore cases of correlated or heterogeneous noise. We note that several steps in the reduction of lensing data can produce correlated pixel noise.
The underfitting bias from the likelihood can presumably be addressed at the expense of an increased statistical error, such as by means of a followup highresolution survey that gathers statistical information on the residuals R_{pre}. First of all, one obvious way to reduce underfitting bias is to choose a template that provides a good fit to galaxy images to reduce the residuals. The optimal template has the profile f_{r}(r) ∝ S_{r}(r) for a preseeing galaxy profile S_{r}(r); this also optimises the signaltonoise ratio of the ellipticity (Bernstein & Jarvis 2002). Even so, realistic galaxies can never be perfectly fit by an elliptical model; there are always fitting residuals R_{pre}. To deal with these residuals, we can imagine a more elaborate Bayesian scheme where R_{pre} are additional nuisance parameters that we marginalise over by using an empirical distribution P(R_{pre}  p) of R_{pre} given p (prior), that is (72)where ℒ(I  p,R_{pre}) is the likelihood of a postseeing image, correctly specified by the postseeing model m(p) = L(Af_{ρ} + R_{pre}) and the noise distribution. The marginalization would then increase the statistical uncertainty in the posterior P_{ϵ}(ϵ  I) in Eq. (45). It is conceivable to measure the prior in a dedicated survey by assessing the distribution of residuals in template fits to high resolution, high S/N images. This approach would be similar to that of BA14 where a (correct) prior on galaxyspecific descriptors is needed. In comparison to BA14, it is noteworthy here that the residuals R_{pre} in the preseeing frame are those of sheared images for which we infer ϵ; no assumptions on the unsheared, sourceplane images have to be made for this prior. All this, however, still has to be tested, and it is not clear how this can be properly implemented and if this marginalization is not the source of another bias.
A consistent likelihood of single images is not sufficient for a lensing analysis as new inconsistencies for ϵ or g can arise in samples of intrinsically different sources through incorrect priors for intrinsic source parameters, such as sizes or centroid positions (prior bias). Take for example our TMP galaxies in Sect. 4 which have a correctly specified likelihood by definition; they are free of underfitting bias. For the two experiments presented in Figs. 3 and 5, the source sample consists of preseeing images with an unknown distribution of intrinsic parameters. The first experiment considers samples with constant ellipticity, the second samples with constant reduced shear. Despite the absent underfitting, noise bias now emerges in both cases if we apply our uniform prior for the nuisance parameters in the marginal posterior of either ellipticity or reduced shear; see TMP data points at ν ≲ 20. We argue in Sect. 3.3 that the emerging noise bias is related to the specific choice of priors and can only be avoided by a special family of correct priors related to the maximum of the posterior of noisefree images. According to previous work, correct priors are the distributions of nuisance parameters of the sources in the sample which have to be defined externally (e.g. BA14, Miller et al. 2013; Kitching et al. 2008). A conceivable alternative to external priors, potentially worthwhile investigating in future studies, might be to specify the priors down to a family of distributions, socalled hyperpriors, with a finite, preferably small number of hyperparameters (Gelman et al. 2013). A hierarchical approach would then jointly determine the posterior density of ϵ or g and the hyperparameters from all I_{i} alone in a fully Bayesian way. Marginalising over the hyperparameters accounts for the uncertainty in the priors. We note that the incorrect uniform prior for ϵ_{s} in Fig. 5 has no significant impact on the shear bias (compare open to filled data points). Clearly, the importance of a correct prior depends on the type of nuisance parameter.
That prior bias in Fig. 5 or Table 1 becomes relevant only at low S/N can be understood qualitatively. For a sufficiently high S/N, the mass of an (identifiable) likelihood ℒ(I_{post}  p) concentrates around p_{post} such that the prior P_{p}(p) details have no strong impact on the marginal posterior of one source. We also expect for high S/N the likelihood to be well approximated by a normal Gaussian with maximum p_{post} such that the marginalization over q = (A,t,x_{0}) approximately yields a Gaussian posterior P_{ϵ}(ϵ  I_{post}) with maximum near ϵ_{post}, which is the true preseeing ellipticity for a correctly specified likelihood. We note that the marginalization over q of a multivariate Gaussian with maximum at p_{0} = (ϵ_{0},q_{0}) produces another Gaussian density with maximum at ϵ_{0}. In addition, should the likelihood be misspecified, ϵ_{post} will be prone to underfitting bias which is clearly visible by the offsets in m for EXP and DEV in Fig. 5.
Finally, we point out that while marginal posteriors P_{ϵ}(ϵ  I_{i}), obtained on an imagebyimage basis, are sufficient to infer a constant ellipticity (or shear) it is not obvious how to incoorporate them into a fully Bayesian analysis of sources with varying shear. For example, in a fully Bayesian inference of ellipticity correlations for images ij at separation ϑ, we would compute the likelihood for given values of ξ_{ϑ} and a range of ϑ, which principally requires the repeated shape measurements for the entire sample of images for every new ξ_{ϑ}. This poses a practical, computational problem. Proposed solutions to this problem leave the path of a fully Bayesian analysis and commonly use unbiased point estimators of ϵ_{i} on a imagebyimage basis as input for a statistical analysis (e.g. Miller et al. 2013). This technique has essentially been applied to all lensing analyses so far, but requires a calibration of the ellipticity estimates, especially for the noise bias. The recently proposed hierarchical Bayesian technique by Alsing et al. (2016), applied to data in Alsing et al. (2017), is closest to a fully Bayesian analysis in the above sense but also uses as input unbiased estimators of source ellipticities with a multivariate Gaussian probability density for the statistical errors of the estimates. Yet, it is surely conceivable to extend this technique by directly modelling the likelihood of source images I_{i}. To reversepropagate our P_{ϵ}(ϵ  I_{i}) to a posterior of ξ_{ϑ}, we might be tempted here to MonteCarlo a posterior PDF of ξ_{ϑ} by independently drawing ellipticities ϵ_{im} ~ P_{ϵ}(ϵ  I_{i}) from the marginal posteriors to produce realisations of a joint vector ϵ_{m} = (ϵ_{1m},...,ϵ_{nm}), which could be used with an estimator of ξ_{ϑ}. Unfortunately, this falsely assumes a separable joint posterior with statistically independent ellipticities, potentially biasing (low) constraints on ξ_{ϑ}. As alternative we speculate about the following hierarchical Bayesian scheme. We assume a parametric model for the joint posterior density of ellipticities that is determined by (i) the measured marginal ellipticities P_{ϵ}(ϵ  I_{i}) and (ii) a set of unknown hyperparameters θ = (θ_{1},θ_{2},...) that quantify the correlations between the source ellipticities. A convenient choice in this respect seem to be copula models that are specified exactly that way (e.g. Sect. 2 in Takeuchi 2010). We further express our ignorance about θ_{i} by a (sufficiently uninformative) hyperprior P_{θ}(θ). We then produce realisations ϵ_{m} by first randomly drawing θ_{m} ~ P_{θ}(θ) followed by . A sample { ϵ_{1},ϵ_{2},... } of joint ellipticities is then utilised to propagate uncertainties on source ellipticities and their mutual correlations through the lensing analysis, based on an estimator for ξ_{ϑ} given a joint sample ϵ. We note that in the limit ν → ∞ the hyperprior for θ becomes irrelevant because all marginal posteriors P_{ϵ}(ϵ  I_{i}) will be sharply peaked at the true ellipticity (if correctly specified).
This assumption would be false if shear and intrinsic ellipticity were correlated, for instance due to selection effects related to the shape of a sheared image (Hirata & Seljak 2003; Heymans et al. 2006). Thus correlations between intrinsic shapes and shear may affect Bayesian methodologies already at this level.
Acknowledgments
The authors are particularly grateful to Gary Bernstein who read and commented on an earlier manuscript of this paper. The comments lead to a substantial improvement. We would also like to thank Tim Schrabback for helpful discussions and comments, and our colleague Reiko Nakajima for organising a weekly seminar on shape measurements in our institute that triggered the research presented in this paper. We also acknowledge the comments by the referee of this paper that helped to clarify and revise certain points, in particular the section on the prior bias. This work has been supported by the Deutsche Forschungsgemeinschaft in the framework of the Collaborative Research Center TR33 “The Dark Universe”. Patrick Simon also acknowledges support from the German Federal Ministry for Economic Affairs and Energy (BMWi) provided via DLR under project No. 50QE1103.
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Appendix A: Leastsquare conditions
In this section, we derive general characteristics of GLAM parameters p = (A,x_{0},M) at a local minimum of the error functional E(p  I), Eq. (3). For convenience, we introduce the bilinear scalar product (A.1)to write the functional as (A.2)Notice that ρ: = (x − x_{0})^{T}M^{1}(x − x_{0}) is a function of p. The scalar product has the properties of which we make heavily use in the following.
Starting from here, we find as necessary condition for the leastsquare solution of any parameter p_{i}: (A.3)or simply (A.4)that is fulfilled for all p_{i}.
Appendix A.1: Template normalisation
From (A.4) follows for the normalisation p_{i} = A at the minimum of E(p  I) that (A.5)
Appendix A.2: Image centroid position
For the leastsquare centroid position x_{0,i} with i = 1,2 at the minimum of E(p  I), we find ∂ [ Af(ρ) ] /∂x_{0,i} = Af′(ρ) ∂ρ/∂x_{0,i} using f′(ρ): = df(ρ) / dρ. This can be expanded further by considering (A.6)Here we have exploited the symmetry [ M^{1} ] _{ij} = [ M^{1} ] _{ji} and introduced the Kronecker symbol . Therefore we find for the ith component of the centroid position (A.7)or, equally, for both centroid components combined (A.8)The scalar product on the lefthandside has to vanish due to symmetry reasons. Therefore using the righthandside, we find at the minimum (A.9)This shows that the centroid x_{0} at the leastsquare point is the firstorder moment x of the image I(x) weighted with the kernel f′(ρ). The weight is thus the derivative f′(ρ) not f(ρ).
Appendix A.3: Image secondorder moments
For the components of M, we consider the leastsquare values of the inverse W_{ij} = [ M^{1} ] _{ij}, hence values W_{ij} for which ∂ [ Af(ρ) ] /∂W_{ij} = Af′(ρ) ∂ρ/∂W_{ij} vanishes. The function ρ can be expressed as (A.10)using X: = (x − x_{0})(x − x_{0})^{T} = X^{T} and the relation tr(ABC) = tr(BCA) for traces of matrices. From this follows that ∂ρ/∂W_{ij} = [ X ] _{ij} = X_{ij}, or (A.11)by employing Eq. (A.4). This set of (four) equations can compactly be written as (A.12)or utilising Eq. (A.5): (A.13)This last relation shows that the bestfitting template f(ρ) has, up to a constant scalar α, the same central secondorder moments M_{I} as the image, In particular, the bestfitting template Af(ρ) has, for a given profile f(s), the same ellipticity ϵ as the image I(x). Moreover, we derive from (A.13) the relation (A.17)The scalar prefactor equals β = − 2 for f(ρ) = e^{− ρ/ 2}. We note that we have on the righthandside f(ρ) in the denominator rather than f′(ρ).
We evolve the expression of β a bit further by changing the variables Δx = M^{1 / 2}y and hence for ρ = Δx^{T}M^{1}Δx in the integrals of the denominator and nominator of β, (A.18)or after another change of variables and , (A.19)where the last step assumes that the profile f(s) vanishes for s → ∞.
Appendix B: Importance sampling of posterior distribution
An overview of our algorithm for sampling the posterior P_{p}(p  I) is given by Algorithm 1, more details follow below.
For the importance sampling of the posterior GLAM variables p, we initially construct an analytical approximation Q(p) of the (full) posterior P_{p}(p  I): = ℒ(I  p) P_{p}(p) in Eq. (45), the socalled importance function. Our importance function is a local Gaussian approximation of ℒ(I  p) that is truncated for  ϵ  ≥ 1; Q(p) is hence defined by its mean and its covariance. As previously discussed, we ignore residuals and set R_{pre} ≡ 0 in Eq. (48) for the following.
For the mean p_{ml} of the Gaussian Q(p), we numerically determine the maximumlikelihood (ML) point of ℒ(I  p) by minimising (48) with respect to p. This is the most time consuming step in our algorithm, which is done in the following three steps.

1.
We quickly guess a starting point p_{0} of the centroid position x_{0} and the galaxy size t by measuring the unweighted first and secondorder moments of the postseeing postage stamp. The initial ellipticity ϵ is set to zero. Herein we ignore the PSF. Alternatively, one could use estimates from object detection software such as SExtractor if available (Bertin & Arnouts 1996).

2.
For an improved initial guess of the size t and the centroid position x_{0}, starting from p_{0} we perform a series of onedimensional (1D) searches wherein we vary one parameter p_{i} while all other parameters p_{j} with i ≠ j are fixed (coordinate descent; Bezdek et al. 1987). For each 1D search, we compute − 2lnℒ(I  p) at five points over the range of values of p_{i} that are allowed according to the prior P_{p}(p). We spline interpolate between the points to estimate the global minimum . Then we update p_{i} to the value and move on to the 1D search of the next parameter. After every update we replace A by its most likely value (B.1)(Eq. A.5) given the current values of (x_{0},ϵ,t). We carry out the searches first for the size t and then for the two components of the centroid position x_{0}.

3.
Starting from p in step 2, we apply the LevenbergMarquardt algorithm (LMA) to iteratively approach p_{ml} (Marquardt 1963). This algorithm devises a linear approximation of Lf_{ρ} at p based on the partial derivatives of Lf_{ρ}. Our implementation of the LMA searches only in the directions of (x_{0},ϵ,t) and again updates A to A_{ml} after each iteration as in step 2. We find a slow convergence for large ellipticities, that is  ϵ  ≳ 0.3. This is improved, however, by adding an extra 1D search for the size t, ϵ_{1} and ϵ_{2} after each LMA iteration with an interval width of ± 0.1t (10 spline points) and ± 0.1 (5 points) around the latest values of t and ϵ_{i}, respectively. For  ϵ_{n}  ≥ 0.6, we use 10 instead of 5 spline points which increases the accuracy for very elliptical objects. We repeat the LMA until iteration n> 5 for which  ϵ_{n − 1} − ϵ_{n}  ≤ 10^{4}; we change this threshold to 2 × 10^{5} once  ϵ_{n}  ≥ 0.6. The latter is needed because the change of ϵ_{n} at each iteration becomes small for high ϵ.
For the covariance of Q(p), we then numerically compute the elements F_{ij} of the (expected) Fisher matrix F around the ML point p_{ml}, (B.2)by considering partial derivatives of the bestfitting GLAM templates (e.g. Tegmark et al. 1997). This gives us (B.3)where H(x) denotes the Heaviside step function. To quickly construct the Fisher matrix, we employ the partial derivatives of Lf_{ρ} in the last iteration of the previous LMA; the derivative with respect to A is simply Lf_{ρ}.
For the main loop of the importance sampling, we randomly draw N_{real} points p_{i} ~ Q(p). Each realisation p_{i} receives an importance weight w_{i} = P_{p}(p_{i}  I) Q(p_{i})^{1} for which we have to evaluate the posterior P_{p}(p_{i}  I); neither P_{p}(p  I) nor Q(p) need to be normalised here. In the end, we normalise all weights to unity for convenience, that is ∑ _{i}w_{i} ≡ 1. Moreover, for later usage, we keep only the ellipticities ϵ_{i} of the parameter sets p_{i}. This Monte Carlo marginalises over the uncertainties of all other parameters.
As another technical detail, the above algorithm has to repeatedly produce pixellated and PSF convolved versions of GLAM templates Lf_{ρ} for a given set of parameters p. To implement a fast computation in this regard, we construct the template images on a high resolution (highres) 256 × 256 grid in Fourier space. To reduce boundary effects, we make the angular extent of the highres grid about 20 per cent larger than the extent of the postseeing galaxy image. Let be the Fourier transform of the PSF and (B.4)
the transform of the radial template profile f(ρ). Then the Fourier coefficients of the PSFconvolved template are (B.5)After setting up the grid of values in Fourier space, we backtransform to coordinate space by means of the FFTW package (Frigo & Johnson 2005). We note that the Fourier coefficient at ℓ = 0 has to be set to to avoid a constant offset of the template. After the FFT, we apply pixellation to produce the vector Lf_{ρ} of postseeing pixel values by averaging pixels of the highres grid within the extent of the postseeing image. We choose the angular extent of the highres grid such that an integer number of highres pixels is enclosed by one postseeing pixel. For example, for our 20 × 20 postseeing images with pixel size r_{post} = 0.103125 ≈ 0.1 arcsec, we use a highres grid with extent of 2.4 arcsec along each axis, or a highres pixel size of r_{hres} = 2.4 / 256 = 0.009375 arcsec (square). Hence exactly (r_{post}/r_{hres})^{2} = 121 highres pixel correspond to one postseeing pixel.
Furthermore, to quickly compute the partial derivatives of ∂(Lf_{ρ}) /∂p_{i}, needed for the LMA and the Fisher matrix, we carry out the foregoing procedure, but we insert the derivatives as Fourier coefficients instead. These can be computed analytically from Eq. (B.5).
All Tables
Multiplicative bias m in per cent for three profiles TMP, EXP, DEV, and sizes r_{h} = 0.2′′ only.
All Figures
Fig. 1
Examples of GLAM templates in the preseeing frame, f_{ρ} (top left), and the postseeing frame, Lf_{ρ} (other panels); the templates are Gaussian radial profiles with f(ρ) = e^{− ρ/ 2}. The bottom left panel simulates only pixellation, whereas the right column also shows the impact of a PSF, indicated in the top right corner, without (top) and with pixellation (bottom). 

In the text 
Fig. 2
Toymodel demonstration of a maximumlikelihood estimator (red), a maximumlikelihood estimator with firstorder bias correction (green), and an estimator exploiting the full posterior (blue). Data points display the estimator average (yaxis) over 10^{6} data points at varying signaltonoise levels (xaxis). The true value to be estimated is x = 1. The panels show different signaltonoise regimes; − nppt denotes y = 1 − n/ 10^{3}. 

In the text 
Fig. 3
Prior bias in the marginal posterior P_{ϵ}(ϵ  I_{1},...,I_{n}) as function of S/N ν for different galaxy sizes r_{h} (in arcsec). The posterior assumes a uniform prior. Shown is the error δϵ =  ϵ − ϵ_{true}  of the inferred ϵ for a true ϵ_{true} = 0.3 as obtained by combining the marginal posteriors of 5 × 10^{3} exposures of the same galaxy with random centroid positions. The pixel size 0.1 arcsec equals the PSF size (Moffat). Galaxy profiles and GLAM templates have a Sérsic profile with n = 2: there is no underfitting. 

In the text 
Fig. 4
Examples of simulated images of galaxies with random ellipticities. Signaltonoise ratios from left to right and top to bottom: ν = 10, 20, 40, and 200. The radial light profile of the preseeing images is exponential with r_{h} = 0.2′′; the pixel size is 0.1′′. The FWHM of the PSF is indicated by the circle in the top right corners. 

In the text 
Fig. 5
Plots of the multiplicative bias m for simulated images, based on Eq. (69), for two different priors P_{s}(ϵ_{s}) (filled data points: correct prior; open data points: uniform prior). Different styles for the data points indicate different image sizes r_{h}, see key inside figure, while colours vary with galaxy type: GLAM template (TMP; red); exponential (EXP; green); bulgelike (DEV; blue). Data points for r_{h} = 0.3′′, same galaxy type, and same prior are connected by dotted lines to guide the eye. The prior for galaxy sizes t, amplitudes A, and centroids x_{0} is uniform giving rise to noisedependent prior bias; the constant offset of EXP and DEV is due to underfitting. A square image pixel has the size 0.1′′, which also equals the PSF size. Results for m for a larger PSF size can be found in Table 1. 

In the text 
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