Latent-normal Gibbs sampler for probit regression ($Z_i\mid \beta \sim N(\beta x_i,1)$)

For binary observations with success probability $\Phi (\beta x_i)$, introduce latent $Z_i\sim N(\beta x_i,1)$ and let the sign determine the observation. Given $\beta$ and the observed sign, each latent variable has a truncated normal distribution. With a standard normal prior, the full conditional distribution of $\beta$ is normal with precision $1+{\sum }_i{x}_i^2$ and mean ${\sum }_i{x}_i{Z}_i/(1+{\sum }_i{x}_i^2)$. Alternating these blocks is Gibbs sampling with the required marginal posterior.