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  • $\begingroup$ It’s worth mentioning, in view of Lindley’s paradox, that your inferences here are going to depend a lot on your prior. The uniform prior makes it really hard to reject when the true probability is close to, but not quite, 1/2, which is probably the case with an actual coin. $\endgroup$ Commented Aug 26, 2018 at 20:14
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    $\begingroup$ True, but it is inherently difficult to distinguish parameter values that are close together, and this naturally requires a lot of data. So I don't really see that as a drawback of the method; it is just a natural aspect of statistics. $\endgroup$ Commented Aug 27, 2018 at 0:37
  • $\begingroup$ It's hard to distinguish points that are close together, sure. But Bayesian inference under a uniform prior diverges drastically from Frequentist inference in this particular case. I can decrease the Bayes factor in favor of the alternative by a factor of 5 just by consider a Uniform(.4,.6) prior under the alternative rather than a Uniform(0,1); this applies in settings where the Uniform(.4,.6) prior and Uniform(0,1) prior result in essentially the same inference about $\pi$ when you don't include a point mass at $1/2$. $\endgroup$ Commented Aug 27, 2018 at 2:11
  • $\begingroup$ Let me see if I can dumb down your answer a bit: We basically comsider not just the probability of experiencing a result for the null hypothesis, we alsp consider the probability of experiencing the result we got given a whole range of heads probabilities from 0 to 1 $\endgroup$ Commented Sep 15, 2018 at 2:59
  • $\begingroup$ @moonman: Yes, that is the essence of Bayesian analysis --- we have an unknown probability of heads, represented by a parameter $\theta$, and we give this a prior distribution and then determine the posterior from the data. In hypothesis testing this generally entails giving a distribution under specific values, not just the general alternative hypothesis. $\endgroup$ Commented Sep 15, 2018 at 3:39