Let X1,..., Xm and Y1,..., Yn be independently distributed as N(, 2) and N(, 2), respectively

Let X1,..., Xm and Y1,..., Yn be independently distributed as N(ξ, σ2) and N(η, τ 2), respectively and let (ξ, η, σ, τ ) have the joint improper prior density π(ξ, η, σ, τ ) dξ dη dσ dτ = dξ dη(1/σ) dσ(1/τ ) dτ . Extend the result of Example 5.7.4 to inferences concerning τ 2/σ2.

Note. The posterior distribution of η − ξ in this case is the so-called Behrens–Fisher distribution. The credible regions for η − ξ obtained from this distribution do not correspond to confidence intervals with fixed coverage probability, and the associated tests of H : η = ξ thus do not have fixed size (which instead depends on τ/σ). From numerical evidence [see Robinson (1976) for a summary of his and earlier results]

it appears that the confidence intervals are conservative, that is, the actual coverage probability always exceeds the nominal one.