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There is evidence from neuroscience studies that we use this type of subjective Bayesian logic unconsciously every day, something similar to intuition. I think the author is using a very strict mathematical definition of Bayes' theorem whereas Silver is using it more colloquially. You are right that popular science can't be held to the same standard as peer reviewed science. I would argue that in science, it might be worthwhile to do a Fisher and a Bayes analysis, but unfortunately only Fisher (or some similar type test) is taken seriously.