Bayesian Statistics: An Introduction, 4th Edition

2.7 Normal variance

2.7.1 A suitable prior for the normal variance

Suppose that we have an n-sample from where the variance is unknown but the mean μ is known. Then clearly

On writing

(remember that μ is known; we shall use a slightly different notation when it is not), we see that

In principle, we might have any form of prior distribution for the variance . However, if we are to be able to deal easily with the posterior distribution (and, e.g. to be able to find HDRs easily from tables), it helps if the posterior distribution is of a ‘nice’ form. This will certainly happen if the prior is of a similar form to the likelihood, namely,

where κ and S0 are suitable constants. For reasons which ...