soreff comments on Probability is in the Mind - Less Wrong
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This seems like it is asking too much of the results of the coin tosses. Given some prior for the probability distribution of biased coins, each toss result updates the probability distribution. Given a prior probability distribution which isn't too extreme (e.g. no zeros in the distribution), after enough toss results, the posterior distribution will narrow towards the observed frequencies of heads and tails.
Yes, at no point is the exact bias known. The distribution doesn't narrow to a delta function with a finite number of observations. So?