Posterior expression for Gibbs sampling

Question

I am trying to estimate parameters of a two dimensional Normal distribution using Gibbs sampling. While it was very easy transform the posterior equation for mean vector to a single dimension normal function for sampling, I am not able to same for sigma(covariance).

Do I need to use the Wishart distribution as prior and then convert the posterior into a single dimensional gamma function ?

Answer 1

The Wishart distribution is the conjugate prior for the likelihood that comes from assuming a normally distributed error term for the linear regression model. You have to assume that either the covariance matrix is an inverted-wishart or the precision matrix (i.e., the inverse of the covariance matrix) is a wishart distribution.

As far as sampling is concerned it is possible to sample directly from the wishart and even from the multivariate normal. Thus, I do not think it is needed to sample from univariate normals or univariate gammas. Look around in the software you are using to see if it has samplers for the multivariate normal and the wishart.