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Cyan comments on Case study: abuse of frequentist statistics - Less Wrong
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I believe you and would like to know some examples for future reference.
The OP is one such -- Bayesians aren't permitted to ignore any part of the data except those which leave the likelihood unchanged. One classic example is that in some problems, a confidence interval procedure can return the whole real line. A mildly less pathological example also concerning a wacky confidence interval is here.