## False confidence, non-additive beliefs, and valid statistical inference. (arXiv:1607.05051v2 [math.ST] UPDATED)

Statistics has made tremendous advances since the times of Fisher, Neyman,
Jeffreys, and others, but the fundamental questions about probability and
inference that puzzled our founding fathers still exist and might even be more
relevant today. To overcome these challenges, I propose to look beyond the two
dominating schools of thought and ask what do scientists need out of
statistics, do the existing frameworks meet these needs, and, if not, how to
fill the void? To the first question, I contend that scientists seek to convert
their data, posited statistical model, etc., into calibrated degrees of belief
about quantities of interest. To the second question, I argue that any
framework that returns additive beliefs, i.e., probabilities, necessarily
suffers from false confidence---certain false hypotheses tend to be assigned
high probability---and, therefore, risks making systematically misleading
conclusions. This reveals the fundamental importance of non-additive beliefs in
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