tl;dr: NHST and Bayesian-style subjective probability do not mix easily.

Another example of this problem: http://slatestarcodex.com/2014/01/25/beware-mass-produced-medical-recommendations/

Does vitamin D reduce all-cause mortality in the elderly? The point-estimates from pretty much all of the various studies are around a 5% reduction in risk of dying for any reason - pretty nontrivial, one would say, no? Yet the results are almost all not 'statistically significant'! So do we follow Rolf and say 'fans of vitamin D ought to update on vitamin D not helping overall'... or do we, applying power considerations about the likelihood of making the hard cutoffs at p<0.05 given the small sample sizes & plausible effect sizes, note that the point-estimates are in favor of the hypothesis? (And how does this interact with two-sided tests - vitamin D could've increased mortality, after all. Positive point-estimates are consistent with vitamin D helping, and less consistent with no effect, and even less consistent with it harming; so why are we supposed to update in favor of no help or harm when we see a positive point-estimate?)

If we accept Rolf's argument, then we'd be in the odd position of, as we read through one non-statistically-significant study after another, decreasing the probability of 'non-zero reduction in mortality'... right up until we get the Autier or Cochrane data summarizing the exact same studies & plug it into a Bayesian meta-analysis like Salvatier did & abruptly flip to '92% chance of non-zero reduction in mortality'.