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Tenoke comments on Open Thread, April 27-May 4, 2014 - Less Wrong
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This is a bit of an exaggeration.
Additionally, you are only talking about the 'sets of statistical tools', where in my experience the bigger disagreement often lies in whether a person accepts that probabilities can be subjective or not; And yes - this does matter.
Can you please give an example of where the possible subjectivity of probabilities matter? I mean this in earnest.
'From my point of view the probability for X is Y, but from his point of view at the time it would've been Z'. (subjective) vs 'The Probability for X is Y' ('objective').
Honestly though, frequentists use subjective probabilities all the time and you can argue that frequentism is just as subjective as bayesinism, so even that disagreement is quite muddy.
Can you be more concrete? When would this matter for two people trying to share a model and make predictions of future events?