5.4 The Behrens–Fisher controversy
5.4.1 The Behrens–Fisher problem from a classical standpoint
As pointed out in Section 2.6 on ‘Highest Density Regions’, in the case of a single normal observation of known variance there is a close relationship between classical results and Bayesian results using a reference prior, which can be summarized in terms of the ‘tilde’ notation by saying that, in classical statistics, results depend on saying that
while Bayesian results depend on saying that
As a result of this, if then the observation x = 5, say, leads to the same interval, , which is regarded as a 95% confidence interval for θ by classical statisticians and as a 95% HDR for θ by Bayesians (at least if they are using a reference prior). It is not hard to see that very similar relationships exist if we have a sample of size n and replace x by , and also when the variance is unknown (provided that the normal distribution is replaced by the t distribution).
There is also no great difficulty ...