The Normal-Inverse-Gamma distribution $(X,\mu,\sigma^2)\sim NIG(\mu_0, \nu, \alpha, \beta)$ is the conjugate prior for the Normal distribution. However, this would correspond to the marginal distribution of $X$ having a Student-$t$ distribution.

Is there a conjugate prior distribution (that gives a distribution on both location [e.g. $\mu$ and $\sigma^2$]) such that the marginal distribution for $X$ will be Normal (as opposed to Student-$t$, in the case of NIG)?

  • $\begingroup$ An easy fix is to choose a Dirac mass prior on $\sigma$. $\endgroup$
    – Xi'an
    Dec 2, 2020 at 9:24


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