Following a question on X Validated, I became aware of the following descriptions of the pros and cons of Bayesian analysis, as perceived by whoever (Tim Arnold?) wrote SAS/STAT(R) 9.2 User’s Guide, Second Edition. I replied more specifically on the point
It [Bayesian inference] provides inferences that are conditional on the data and are exact, without reliance on asymptotic approximation. Small sample inference proceeds in the same manner as if one had a large sample. Bayesian analysis also can estimate any functions of parameters directly, without using the “plug-in” method (a way to estimate functionals by plugging the estimated parameters in the functionals).
which I find utterly confusing and not particularly relevant. The other points in the list are more traditional, except for this one
It provides interpretable answers, such as “the true parameter θ has a probability of 0.95 of falling in a 95% credible interval.”
that I find somewhat unappealing in that the 95% probability has only relevance wrt to the resulting posterior, hence has no absolute (and definitely no frequentist) meaning. The criticisms of the prior selection
It does not tell you how to select a prior. There is no correct way to choose a prior. Bayesian inferences require skills to translate subjective prior beliefs into a mathematically formulated prior. If you do not proceed with caution, you can generate misleading results.
It can produce posterior distributions that are heavily influenced by the priors. From a practical point of view, it might sometimes be difficult to convince subject matter experts who do not agree with the validity of the chosen prior.
are traditional but nonetheless irksome. Once acknowledged there is no correct or true prior, it follows naturally that the resulting inference will depend on the choice of the prior and has to be understood conditional on the prior, which is why the credible interval has for instance an epistemic rather than frequentist interpretation. There is also little reason for trying to convince a fellow Bayesian statistician about one’s prior. Everything is conditional on the chosen prior and I see less and less why this should be an issue.
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