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

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while Bayesian results depend on saying that

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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 ...

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