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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

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  • $\begingroup$ Is this a self-study question? $\endgroup$ – iliasfl Aug 4 '14 at 10:23
  • $\begingroup$ kind of. I am doing a project where I am facing this problem $\endgroup$ – Neel Aug 4 '14 at 15:24
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    $\begingroup$ 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. $\endgroup$ – guy Aug 4 '14 at 16:42

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