Design by elimination vs. design by comparison
Behind this question are two fundamentally different approaches about how to reason with chance hypotheses. One approach, due to Ronald Fisher, rejects a chance hypothesis provided sample data appear in a prespecified rejection region. The other, due to Thomas Bayes, rejects a chance hypothesis provided an alternative hypothesis confers a bigger probability on the data in question than the original hypothesis. In the Fisherian approach, chance hypotheses are rejected in isolation for rendering data too improbable. In the Bayesian approach, chance hypotheses are eliminated provided some other hypotheses render the data more probable. Whereas in the Fisherian approach the emphasis is on elimination, in the Bayesian approach the emphasis is on comparison. These approaches are incompatible, and the statistical community itself is deeply riven over which of these approaches to adopt as the right canon for statistical rationality. The difference reflects a deep divergence in fundamental intuitions about the nature of statistical rationality and in particular about what counts as statistical evidence.
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