Gaussian-noise posterior for an autologistic image ($p_v^{\mathrm{post}}=\mathrm{logistic}(\alpha +\beta z_v+y_v-\frac{1}{2})$)

Independent unit-variance Gaussian observations with mean $x_v$ add $y_v-1/2$ to the local binary field. The posterior distribution conditional probability is $\mathrm{logistic}(\alpha +\beta z_v+y_v-1/2)$. The neighbor interaction is unchanged, so checkerboard conditional independence and the same Gibbs schemes remain valid.