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I am deriving a gibbs sampler for a joint distribution, where the conditionals of various parameters are product of two non-standard normal distributions. Usually, I have seen that in Gibbs sampling people sample their conditionals from standard distribution. But if I sample from the two normals and then multiply the samples should I be fine or should I use Metropolis Hastings step inside Gibbs?

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  • $\begingroup$ Not sure what you mean by "non-standard" in this context; can you edit the question to include the expressions? The product of normal distributions is again a gaussian function. The product of samples from two normal distributions does not follow a normal distribution law. $\endgroup$ – Cyan Apr 5 '13 at 3:49
  • $\begingroup$ Not standard means not (mean =0 sd=1). Product of two standard normals follows product normal. $\endgroup$ – Blade Runner Apr 5 '13 at 3:56
  • $\begingroup$ In Gibbs sampling, if the conditional distribution is non-standard normal, then that's what gets sampled. It's not clear to me exactly what "the conditionals of various parameters are product of two non-standard normal distributions" means (post the math!), but if it means that the conditional density is the product of two non-standard normal densities, then you need the following product: $\exp\bigg(-\frac{\tau_1}{2}(x-\mu_1)^2\bigg) \cdot \exp\bigg(-\frac{\tau_2}{2}(x-\mu_2)^2\bigg) \propto \exp\bigg(-\frac{\tau_1 + \tau_2}{2}(x-\frac{\tau_1\mu_1 + \tau_2\mu_2}{\tau_1 + \tau_2})^2\bigg)$ $\endgroup$ – Cyan Apr 5 '13 at 4:54

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