Bayesian Statistics: An Introduction, 4th Edition

5.3 Variances unknown and unequal (Behrens–Fisher problem)

5.3.1 Formulation of the problem

In this section, we are concerned with the most general case of the problem of two normal samples, where neither the means nor the variances are assumed equal. Consequently we have independent vectors and such that

and . This is known as the Behrens–Fisher problem (or sometimes as the Behrens problem or the Fisher–Behrens problem).

It is convenient to use the notation of the previous section, except that sometimes we write and instead of and to avoid using sub-subscripts. In addition, it is useful to define

For the moment, we shall assume independent reference priors uniform in λ, μ, and ψ. Then, ...