# Is this summary about Bayesian inference for Normal distribution right?

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

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?

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.

• 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. – DuFei Nov 3 '17 at 1:26
• 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. – jaradniemi Nov 3 '17 at 14:22
• Feel free to mark it as the answer. – jaradniemi Nov 8 '17 at 2:22