Bayesian Statistics: An Introduction, 4th Edition

2.3 Several normal observations with a normal prior

2.3.1 Posterior distribution

We can generalize the situation in the previous section by supposing that a priori

but that instead of having just one observation we have n independent observations such that

We sometimes refer to X as an n-sample from . Then

Proceeding just as we did in Section 2.3 when we had only one observation, we see that the posterior distribution is

where

We could alternatively write these formulae as

which shows that, assuming a normal prior and likelihood, the result is just the same as the posterior distribution obtained from the single observation of the mean , since we know that

and the above formulae are the ...