How are design hypotheses properly inferred - simply by eliminating chance hypotheses or by comparing the likelihood of chance and design hypotheses?
Behind this question are two fundamentally different approaches about how to reason with chance hypotheses, one friendly to intelligent design, the other less so. The friendly approach, due to Ronald Fisher, rejects a chance hypothesis provided sample data appear in a prespecified rejection region. The less friendly approach, 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.
The most influential criticism of specified complexity charges it with falling on the wrong side of this divide. Specifically, critics charge that to use specified complexity to infer design presupposes an eliminative, Fisherian approach to reasoning with chance hypotheses whereas the right approach to inferring design needs to embrace a comparative, Bayesian approach. The most prominent scholar to make this criticism is Elliott Sober. Other scholars have offered this criticism as well, and many more still have cited it as decisively refuting specified complexity as a sign of intelligence.
In responding to this criticism, let's begin with a reality check.
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