| No, but let me quote Jaynes (Probability Theory: the Logic of Science, 10.10): > ‘When I toss a coin, the probability for heads is one-half.’ [...] the issue is between the following two
interpretations: > (A) ‘The available information gives me no reason to expect heads rather than tails, or vice versa – I
am completely unable to predict which it will be.’ > (B) ‘If I toss the coin a very large number of times, in the long run heads will occur about half the
time – in other words, the frequency of heads will approach 1/2.’ These are not the same except for special circumstances like controlled experiments. Frequentism usually assumes or restricts itself to those special circumstances. The long run here means `for any ε > 0, the probability that the
observed frequency n/N lies in the interval (1/2±ε) goes to 1 as N goes to infinity'. There is no such thing as an intrinsic probability, a coin only has a chance of landing heads when tossed. If we know everything about the coin and how it is tossed we could calculate the result. Some ways of tossing a fair coin are biased. Some coins are biased when tossed in fair ways. A fair coin toss exaggerates factors that are difficult to know and control exactly, like the force that we apply on the coin with our finger. Jaynes thinks that `true randomness' such as is postulated by conventional quantum physics is unscientific (10.7). In any case it doesn't matter for calculating, and I don't think many Bayesians lose sleep over whether it's unknown, unknowable or `true randomness'. If the Bayesian has an informative prior, they won't be mislead too much. If a Bayesian is 100% sure beforehand with a δ(x - 0.5) prior they won't be mislead at all, but of course no one is ever 100% sure (Cromwell's rule). On the other hand a frequentist might say p<0.05 and be mislead. |