On some counter-counter-examples about classical inference
AbstractThis paper deals with theoretical concepts and practical examples, aimed at showing that non-Bayesian inference is liable to result in mistakes or unacceptable conclusions, and proves that they are not justified. Section 2 comments on examples when an objective prior distribution exists, and shows how widely one can be mistaken in using a prior quite distant from the real one. Section 3 comments on two results by Godambe, stressing that – in sampling from finite populations – no flat likelihood exists, while an unbiased linear “estimator” with zero variance does not exist, unless we reach a complete knowledge of the population. Section 4 stresses the fundamental difference between a “probability interval” for a parameter, and a “confidence interval” aimed at making inference on the parameter, thus summarizing all certain facts and constraints able to shrink such an inferential interval. Section 5 explains why we are justified in attaching an inductive meaning to a realized confidence interval. Finally, Section 6 counters some well known counter-examples spread in the Bayesian literature, showing that they are unacceptable from a sound inductive basis.
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