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Bayesian Lensing Shear Measurement

We derive an estimator of weak gravitational lensing shear from background galaxy images that avoids noise-induced biases through a rigorous Bayesian treatment of the measurement. The derived shear estimator disposes with the assignment of ellipticities to individual galaxies that is typical of previous approaches to galaxy lensing. Shear estimates from the mean of the Bayesian posterior are unbiased in the limit of large number of background galaxies, regardless of the noise level on individual galaxies. The Bayesian formalism requires a prior describing the (noiseless) distribution of the target galaxy population over some parameter space; this prior can be constructed from low-noise images of a subsample of the target population, attainable from long integrations of a fraction of the survey field. We find two ways to combine this exact treatment of noise with rigorous treatment of the effects of the instrumental point-spread function and sampling. The Bayesian model fitting (BMF) method assigns a likelihood of the pixel data to galaxy models (e.g. Sersic ellipses). The Bayesian Fourier domain (BFD) method compresses the pixel data to a small set of weighted moments calculated after PSF correction in Fourier space. A numerical test using a simplified model of a biased galaxy measurement process demonstrates that the Bayesian formalism recovers applied shears to $<1$ part in $10^3$ accuracy and provides accurate uncertainty estimates. BFD is the first shear measurement algorithm that is model-free and requires no approximations or ad hoc assumptions in correcting for the effects of PSF, noise, or sampling on the galaxy images. These algorithms are good candidates for attaining the part-per-thousand shear inference required for hemisphere-scale weak gravitational lensing surveys. (abridged)