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I am studying conditional probability, and I came across this Bayes theorem. Though I am no statistician, I kinda get the idea of the theorem, that probabilities can be updated using newly given informations.

What I am curious about is: I heard that Bayesian statistics was a "new" approach to statistics, and therefore was met with many objections and criticisms. But I learned that the Bayesian theorem $$P(B|A)=\frac{P(A|B)P(B)}{P(A)}$$ can be derived from the definition of conditional probability. I can't understand why this was met with criticisms. I heard that they were due to different views about the meaning of probability and inference (like frequentists and propensitists), but I don't really see any difference between these.

Would someone please explain how Bayesians were at odds with other views about statistics, and why Bayesian approach aroused numerous arguments and confusions (with some examples if possible)?

Now my question might sound a bit vague, but I have almost no background information about advanced statistics, so this is the best I can give. Thank you in advance!

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  • $\begingroup$ Baynesian methods introduce some very subjective elements (especially the choice of the prior) which can be undesirable. This paper goes into more detail. $\endgroup$ – Mark Feb 6 '17 at 1:19
  • $\begingroup$ This article is much more easily read. An important takeaway is that Bayes' Theorem itself is entirely uncontroversial, just that the Baynesian interpretation of it (in updating a prior) can be controversial in certain settings. $\endgroup$ – Mark Feb 6 '17 at 1:23
  • $\begingroup$ Another concept is that Bayesian methods where the parameter comes from some true distribution can have some frequentist properties. For example if $X \sim \textrm{N}(\mu,1)$ with $\mu$ some true values. You might want frequentist properties of the posterior, in the sense that asymptotically you get a posterior which contracts to the true value of $\mu$ independent of what prior you specify. The Bernstein Von Mises Theorem is a good example of this. $\endgroup$ – Ryan Warnick Mar 5 '18 at 7:29
  • $\begingroup$ These notes are very technical, but give examples of this in more abstracted settings on Chapter 7; including things like infinite dimensional priors. $\endgroup$ – Ryan Warnick Mar 5 '18 at 7:30

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