8.5 Bayesian analysis for an unknown overall mean
In Section 8.2, we derived the posterior for 
supposing that a priori
where μ was known. We shall now go on to an approach introduced by Lindley (1969) and developed in his contribution to Godambe and Sprott (1971) and in Lindley and Smith (1972) for the case where μ is unknown.
We suppose that
are independent given the 
and 
. This is the situation which arises in one way analysis of variance (analysis of variance between and within groups). In either of the practical circumstances described above, the means 
will be thought to be alike. More specifically, the joint distribution of these means must have the property referred to by de Finetti (1937 or 1974–1975, Section 11.4) as exchangeability; that is, the joint distribution remains invariant under any permutation of the suffices. A famous result in de Finetti (1937) [for a good outline ...
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