For the multivariate normal model, why is jeffreys' prior distribution not a probability density?

Question

For the multivariate normal model, Jeffreys' rule for generating a prior distribution on (theta, sigma) gives p_j(theta, sigma) proportional to |sigma|^{-(p+2)/2}.

My book notes in a footnote that p_j cannot actually be a probability density for theta, sigma. Why is this?

Answer 1

It's "improper", meaning it doesn't integrate to 1 as probability distributions have to do. For example, the marginal density with respect to theta is just a constant, whose integral over the real line is infinite. It's OK to use improper distributions as priors in Bayesian inference, as long as the posterior is a proper probability distribution.