Let $X \sim \mathcal{N}(M, S)$, where $M,S$ are themselves gaussian random variable with mean $\mu_{M,S}$ and variance $\sigma_{M,S}$. Does this distribution have a particular name/form? Can we compute its CDF?
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1How is $\mathcal{N}(M,S)$ defined when $S$ is negative? – Stefan Hansen Jan 24 '14 at 15:37
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If S is Gaussian then it could be negative which does not work. This problem is common in Bayesian statistics. Possibly specifying the precession $=1/S^2$ will work better. It will be easy to integrate out the M, this will just give a new Gaussian where the mean depends on $\mu_M$, $S$ and $\sigma_M$. Integrating out the S will be messy and possibly not possible.
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