In this paper, we extend the Bayesian model fitting shape measurement method presented in Miller et al., and use the method to estimate the shear from the Shear TEsting Programme simulations (STEP). The method uses a fast model fitting algorithm that uses realistic galaxy profiles and analytically marginalizes over the position and amplitude of the model by doing the model fitting in Fourier space. This is used to find the full posterior probability in ellipticity. The shear is then estimated in a Bayesian way from this posterior probability surface. The Bayesian estimation allows measurement bias arising from the presence of random noise to be removed. In this paper, we introduce an iterative algorithm that can be used to estimate the intrinsic ellipticity prior and show that this is accurate and stable.

We present results using the STEP parametrization that relates the input shear γ^T to the estimated shear γ^M by introducing a bias m and an offset c: γ^M - γ^T = mγ^T + c. The average number density of galaxies used in the STEP1 analysis was 9 per square arcminute, for STEP2 the number density was 30 per square arcminute. By using the method to estimate the shear from the STEP1 simulations we find the method to have a shear bias of m = 0.006 +/- 0.005 and a variation in shear offset with point spread function type of σ_c = 0.0002. Using the method to estimate the shear from the STEP2 simulations we find that the shear bias and offset are m = 0.002 +/- 0.016 and c = -0.0007 +/- 0.0006, respectively. In addition, we find that the bias and offset are stable to changes in the magnitude and size of the galaxies. Such biases should yield any cosmological constraints from future weak lensing surveys robust to systematic effects in shape measurement.

Finally, we present an alternative to the STEP parametrization by using a quality factor that relates the intrinsic shear variance in a simulation to the variance in shear that is measured and show that the method presented has an average of Q >~ 100 which is at least a factor of 10 times better than other shape measurement methods.