4.4 Point (or sharp) null hypotheses with prior information

4.4.1 When are point null hypotheses reasonable?

As was mentioned in Section 4.3, it is very common in classical statistics to conduct a test of a point (or sharp) null hypothesis

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In such a case, the full-scale Bayesian approach (as opposed to the compromise described in the previous section) gives rise to conclusions which differ radically from the classical answers.

Before getting on to the answers, a few basic comments about the whole problem are in order. First, tests of point null hypotheses are often performed in inappropriate circumstances. It will virtually never be the case that one seriously entertains the hypothesis that  exactly, a point which classical statisticians fully admit (cf. Lehmann, 1986, Sections 4.5, 5.2). More reasonable would be the null hypothesis

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where  is so chosen that all  can be considered ‘indistinguishable’ from . An example in which this might arise would be an attempt to analyze a chemical by ...

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