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I've heard of the one found here:

http://www.behind-the-enemy-lines.com/2008/01/are-you-bayesian-or-frequentist-or.html

about flipping a coin fourteen times, having it come up heads ten times, and then betting on whether the coin would come up heads on the next two tosses.

Are there others? Are there any really classic problems in this area?

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  • $\begingroup$ From your link: "If we assume that we know nothing about p, we can assume that the prior is a uniform distribution" — is wrong and so his Bayesian calculation is also wrong. But his essay does make a case that frequentism is just Bayesian reasoning with an unstated prior. $\endgroup$
    – Neil G
    Jun 2, 2015 at 23:22
  • $\begingroup$ @NeilG, is "wrong" not rather a strong statement? Of course, also widely discussed here on CV, a flat prior is not the same as one representing lack of knowledge, but how can it be called "wrong" to pick such a prior? $\endgroup$
    – Christoph Hanck
    Jun 3, 2015 at 3:39
  • $\begingroup$ @Hanck: It's wrong to call it uninformative since it assumes a parametrization. If it were right, then we arrive at contradiction because I can assume a flat prior for the same problem using a different parametrization, which is different. $\endgroup$
    – Neil G
    Jun 3, 2015 at 3:51

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Bayesian and frequentist approaches give different answers in every problem since the interpretation of the two paradigms is different. In point estimation, Bayesians provide maximum a posterior (MAP) estimates and frequentist provides something else, e.g. maximum likelihood estimates (MLE) or method of moments. In interval estimation, Bayesians provide credible intervals (CrIs) while frequentists provide confidence intervals (CIs). In hypothesis testing, Bayesians provide posterior probabilities (or Bayes' factors) while frequentists provide pvalues. In each of these areas of statistic, Bayesians and frequentist differ in the answer they give.

Now, there are some times when the numerical results coincide, e.g. MAP=MLE and CrI=CI. Generally, this will only be true when certain improper prior distributions are used. There are also scenarios where hypothesis testing results in the same decision under a Bayesian and frequentist paradigm, but there are definitely times when they don't, e.g. Jeffrey-Lindley Paradox.

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  • $\begingroup$ Are there any problems on which the two methods will give different actual results? Like, bet or don't bet? $\endgroup$
    – user77463
    Jun 4, 2015 at 3:37

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