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I summarize this based on http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf

However, there are some questions about this.

  1. When the mean and variance are both unknown, can I set the prior of of variance a Chi-square distribution? If so, is the posterior normal chi-square distribution?
  2. I know that wishart distribution is generalization of chi-square distribution, then are there any generalizations of gamma distribution?
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No. Only lines 1 and 6 are correct.

For a univariate normal,

  • the conjugate prior for a variance (with known mean) is an inverse gamma (which is a reparameterization of a scaled-inverse $\chi^2$ distribution and thus the inverse $\chi^2$ is a special case) and
  • the conjugate prior for mean and variance is a normal-inverse-gamma distribution.

For a multivariate normal,

  • the conjugate prior for a covariance matrix (with known mean) is an inverse Wishart (which is a generalization of an inverse-gamma distribution) and
  • the conjugate prior for the mean and variance is a normal-inverse-Wishart distribution.

Note that you do not need to specify the posterior family once you have specified the prior family because by definition of conjugacy, these will be the same.

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  • $\begingroup$ thanks very much. Based on your answers, I think lines 2 and 3 are incorrect, lines 4 and 5 are correct, right? For multivariate normal, lines 7 and 8 are incorrect, I will remove them. $\endgroup$
    – DuFei
    Commented Nov 3, 2017 at 1:26
  • $\begingroup$ The way you have laid it out, lines 4, 5, and 9 are not correct because you need a joint distribution for the mean and (co-)variance in both the prior and the posterior. You have indicated this in the posterior, but have not indicated this in the prior. $\endgroup$
    – jaradniemi
    Commented Nov 3, 2017 at 14:22
  • $\begingroup$ Feel free to mark it as the answer. $\endgroup$
    – jaradniemi
    Commented Nov 8, 2017 at 2:22

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