I need some help in finding out precision matrix of a multivariate Gaussian distribution using Gibbs Sampling with a known mean. I tried generating samples using Wishart distribution as a prior distribution, but not able to get the correct output.
I derived the following expression for the scale matrix and the degrees of freedom that I used in the posterior distribution(Wishart distribution).
I am not sure whether these expression is correct or not.
Do I need to sample each component of the precision separately( by rearranging the multivariate expression into univariate expression), or the sampling from the wishart distribution will take care of it ?
For some reason what your derivation isn't showing up on my browser. Wikipedia has the parameters for the conjugate update: conjugate priors. Alexandre is right that the conjugate prior for the covariance matrix is the inverse Wishart (or the Wishart is the conjugate prior for the inverse covariance matrix). Essentially the updated covariance parameter to the inverse-Wishart distribution is the sum over the prior covariance matrix, outer product of each data point (centered at the sample mean), and the outer product of the sample mean (centered at the prior mean) weighted by the k0*N/(k0+N),
where k0 is the prior variance on the mean, and N is the number of data points.
The updated degrees of freedom of the inverse wishart is N + v0. (where v0 is the prior degrees of freedom). I think v0 has to be >= the dimensionality of the data, but i forget.
answered Nov 27 '10 at 14:14