Re: How do we choose covariance matrix for multivariate normal?

On Jul 17, 7:43 am, Maniaoh <n.hoai...@xxxxxxxxx> wrote:
Hi there,

I have a question about Bayesian inference and a little bit related to
Markov chain Monte Carlo. Suppose that I have observed data D and I
want to have a model describing that data. I may test it by using
model f(x, theta) where theta = (a, b, c) (a vector with some
elements). We assume that (a, b, c) is multivariate normal so making
inference about them requires covariance matrix C, which is used in
MCMC method. The problem is that I do not know how C is built or
chosen, given the observed data D.

Please give me some instructions on this matter or refer me to any
related documents. Thank you very much in advance.

Iaoh.

Since you're trying to be Bayesian, go ahead and treat the covariance
matrix as unknown and use a Normal-Wishart distribution to model your
data. This is a multivariate extension of the scalar problem of
estimating the parameters of a normal distribution with an unknown
mean and covariance.

Any upper-level/graduate textbook on Bayesian data analysis should
discuss this model. Bayesian Data Analysis by Gelman, Carlin, Stern
and Rubin is a good place to start.

Here is an early journal paper on the problem, too.

Bayesian Analysis of the Independent Multinormal Process. Neither Mean
Nor Precision Known by Albert Ando and G. M. Kaufman. Journal of the
American Statistical Association, Vol. 60, No. 309 (Mar., 1965), pp.
347-358

-Lucas
.

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