In this paper we extend the Bayesian model fitting shape measurement method presented in Miller et al .
( 2007 ) and use the method to estimate the shear from the Shear TEsting Programme simulations ( STEP ) .
The method uses a fast model fitting algorithm which uses realistic galaxy profiles and analytically marginalises 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 parameterisation which relates the input shear \gamma ^ { T } to the estimated shear \gamma ^ { M } by introducing a bias m and an offset c : \gamma ^ { M } - \gamma ^ { T } = m \gamma ^ { T } + c .
By using the method to estimate the shear from the STEP1 simulations we find the method to have a shear bias of m \sim 5 \times 10 ^ { -3 } and a variation in shear offset with PSF type of \sigma _ { c } \sim 2 \times 10 ^ { -4 } .
These values are smaller than for any method presented in the STEP1 publication that behaves linearly with shear .
Using the method to estimate the shear from the STEP2 simulations we find than the shear bias and offset are m \sim 2 \times 10 ^ { -3 } and c \sim - 7 \times 10 ^ { -4 } respectively .
In addition we find that the bias and offset are stable to changes in 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 parameterisation 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 \mathrel { \raise 1.16 pt \hbox { $ > $ } \kern - 7.0 pt \lower 3.06 pt \hbox { { $ \scriptstyle% \sim$ } } } 100 which is at least a factor of 10 times better than other shape measurement methods .