Bayesian Galaxy Shape Measurement for Weak Lensing Surveys -I. Methodology and a Fast Fitting Algorithm

TL;DR

The paper argues that a Bayesian, model-fitting approach to galaxy shape measurement can achieve optimal weak-lensing shear estimates with proper error handling, potentially eliminating external calibration. It introduces a fast algorithm that marginalizes over uninteresting parameters in Fourier space, enabling scalable analysis for large surveys. The method formalizes shear estimation via shear sensitivity and demonstrates robust performance on STEP simulations, highlighting speed, applicability, and avenues for refinement. Together, these elements provide a framework for accurate, scalable, and calibration-free weak-lensing analyses across current and future surveys.

Abstract

The principles of measuring the shapes of galaxies by a model-fitting approach are discussed in the context of shape-measurement for surveys of weak gravitational lensing. It is argued that such an approach should be optimal, allowing measurement with maximal signal-to-noise, coupled with estimation of measurement errors. The distinction between likelihood-based and Bayesian methods is discussed. Systematic biases in the Bayesian method may be evaluated as part of the fitting process, and overall such an approach should yield unbiased shear estimation without requiring external calibration from simulations. The principal disadvantage of model-fitting for large surveys is the computational time required, but here an algorithm is presented that enables large surveys to be analysed in feasible computation times. The method and algorithm is tested on simulated galaxies from the Shear TEsting Program (STEP).

Figures (4)

• Figure 1: Illustration of the properties of an ideal likelihood estimator $\hat{x}_{\mathcal{L}}$ (left) and ideal Bayesian estimator $\hat{x}_{\mathcal{B}}$ (right) for the Gaussian example described in the text. The top pair of graphs show the correlation between the input and deduced values. Two regression lines are shown on each, one being the regression of input on estimated value, the other being the regression of estimated on input value. The next two pairs show the distribution of the difference between input and estimated values compared with either the estimated values (centre) or the input values (bottom). Note the graph x-axes differ between the centre and bottom panels. For a given input value, the likelihood estimator yields an unbiased estimate (regression slope unity) whereas the Bayesian estimator appears biased (regression slope 2.75). However, for a given estimated value, the likelihood estimator is biased (regression slope 0.36) and the Bayesian estimator is unbiased. The Bayesian estimator returns the best estimate of the input value for a given measurement.
• Figure 2: Comparison of Bayesian posterior probability $p(\bmath{e})$ (left) and likelihood $\mathcal{L}(\bmath{e})$ (right) surfaces for two individual galaxies. The grey-scale is logarithmic showing a range of 5 in $\Delta\log\mathcal{L}$ below the maximum value (shown as white) in each case. The upper panel shows results from fitting to a magnitude $24.17$ simulated STEP galaxy, the lower panel a magnitude $23.15$ galaxy. Solid lines show the two parameter $1$-$\sigma$ and $2$-$\sigma$ contours. The cross shows the input ellipticity value.
• Figure 3: Tests on the STEP 1 simulated galaxy sample, as a function of galaxy apparent magnitude. Each graph shows the expectation value of the Bayesian estimate of component $e_1$ (x-axis) plotted against the input value (y-axis). Results for component $e_2$ are similar and are not shown. Left-hand panels show individual simulated galaxies, right-hand panels show results binned in intervals of the measured ellipticity. Two magnitude ranges are shown, $m > 22$ (upper panels) and $m \le 22$ (lower panels). The solid lines have a slope of unity, the dashed lines on the left-hand panels show the least-squares regression of input values on estimated values. The mean error on individual measured ellipticities is shown on the left-hand panels. Vertical error bars on the right-hand panels indicate the error in the mean input values in each interval of measured values.
• Figure 4: The summed posterior probability distribution of measured ellipticity values $e_1$ (top) and $e_2$ (bottom) as a function of apparent magnitude. The prior $\mathcal{P}(\bmath{e})$ is also shown for comparison as a dashed line. The magnitude ranges of the simulated galaxies are $m\leq 22$ (left panels) and $m>22$ (right panels).