No, I'm not making this up - people who prefer the (old-fashioned? classic?) non-Bayesian kind of probability analysis really do call themselves 'frequentists'.
Both kinds of analysis deal with the general problem that we can't absolutely know the truth, but must use samples and/or tests to approximate it. Here's what I think the differences are:
The classic approach thinks in terms of samples of the real world, and calculates how closely the samples are likely to reflect reality. If one sample from a bacterial culture, plated on novobiocin agar, gives 28 (NovR) colonies, and a parallel sample (same volume) on plain agar gives 386 colonies, we can calculate the probability that the whole culture has 28/386 NovR cells. We can use replicate samples to estimate the error in, for example, measuring the volumes of culture we used.
The Bayesian approach thinks in terms of the reliability of our information and tests, and calculates how the result of a test changes our previous estimate about reality. For example, based on previous similar experiments we might have estimated the NovR frequency at 15%. But we also know that the plating test isn't perfect, and we can estimate how likely it is to be wrong in different ways. Depending on our expertise, we might take into account the risk that NovS colonies grow on novobiocin agar or fail to grow on plain agar, and how much error our volume measurements have. Bayes' theorem and the associated methods tell us how to revise our original estimate (15% NovR) in the light of the new information (28/386 NovR).
OK, I think this is as far into Bayesian analysis as I want to go. But I'd greatly appreciate comments from readers with more expertise, telling me if what I've written in these three posts is seriously off track.
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