I am performing a Bayesian multivariate regression, and therefore I have to construct the prior and the subsequent posterior.
But the paper that I am using as a reference, uses a "Uninformative prior", and moreover, gives me directly the expression of the posterior.
So what I did, was just drawing from the posterior directly, without considering the prior at all.
A detail of what I meant is this:
A standard uninformative prior here is
$$ p(C, \Sigma) \propto |\Sigma|^{-(n+2)/2} $$
the posterior $p(C, \Sigma^{-1}\mid z)$ is given by
$$ \Sigma^{-1}\mid z \sim \mathrm{Wishart}(T-n-2, S^{-1}) \\
vec(C)\mid \Sigma,z \sim N(vec(\hat C), (X'X)^{-1}) $$
I just draw from the posterior (so the last two expressions).
I didn't understood the meaning of uninformative prior, and if, since it gives me no information, I can "skip it" and use directly the posterior derived from it.
Let me know if my reasoning is correct, or if you can suggest me how to incorporate the prior (how is that distribution defined)?
Best Answer
By Bayes theorem
$$ \text{posterior} \propto \text{prior} \times \text{likelihood} $$
so posterior combines information that comes from your data (through likelihood) and information that comes from your prior. So there is no posterior without prior. In case of the quote you provided, it provides a closed-form posterior distribution, because it uses the fact that Wishart distribution is a conjugate prior for precision parameter of multivariate normal distribution, so we have a "standalone" closed-form for it.
This means that for such model you do not have to sample from the prior distribution to conduct a MCMC simulation, since the posterior is known.
This does not mean that we do not use prior in here, we have already used it. On another hand, if you do not want to use prior, then you should rather use maximum likelihood estimation rather then Bayesian estimation.