# Conjugate Prior for Probit likelihood function

I am trying to do a Bayesian analysis in which my likelihood function is a probit function on two parameters. From various sources, I found out that Normal distribution is a conjugate prior to probit likelihood, but I am not being able to compute the posterior from the functions. Here are the details of the problem

Likelihood: $\Phi(\theta-\beta)$

Prior for $\beta$: $\phi(\beta-\mu_1)$

Prior for $\theta$: $\phi(\theta-\mu_2)$

Joint Posterior: $C\Phi(\theta-\beta)\phi(\beta-\mu_1)\phi(\theta-\mu_2)$

Now I want to get the marginal distributions from this joint posterior

• Is this a self-study question? – iliasfl Aug 4 '14 at 10:23
• kind of. I am doing a project where I am facing this problem – Neel Aug 4 '14 at 15:24
• The normal is not conjugate, but it is conditionally conjugate if you introduce a latent Gaussian $Z$ Such that $Z > 0$ corresponds to a success. To use this idea for Bayesian inference, one then runs a Gibbs sampler. – guy Aug 4 '14 at 16:42