What are the cons of Bayesian analysis? What are some practical objections to the use of Bayesian statistical methods in any context? No, I don't mean the usual carping about choice of prior. I'll be delighted if this gets no answers.
 A: Sometimes there's a simple and natural "classical" solution to a problem, in which case a fancy Bayesian method (especially with MCMC) would be overkill.  
Further, in variable selection type problems, it can be more straightforward and clear to consider something like a penalized likelihood; there may exist a prior on models that gives an equivalent Bayesian approach, but how the prior corresponds to the ultimate performance can be less clear than the relationship between the penalty and performance.
Finally, MCMC methods often require an expert both for assessing convergence/mixing and for making sense of the results.
A: I am relatively new to Bayesian methods, but one thing that that irks me is that, while I understand the rationale of priors (i.e. science is a cumulative endeavour, so for most questions there is some amount of previous experience/thinking that should inform your interpretation of the data), I dislike that the Bayesian approach forces you to push subjectivity to the beginning of the analysis, rendering the end result contingent. I believe this is problematic for two reasons: 1) some less well versed readers won't even pay attention to the priors, and interpret Bayesian results as non-contingent; 2) unless the raw data is available, it is hard for readers to reframe the results in their own subjective priors. This is why I prefer likelihood ratios, they put the subjectivity at the end by simply quantifying relative evidence values and leaving it up to the reader to apply their own subjective criteria to determine whether they believe the evidence is sufficiently in favor of either model involved in the ratio.
(Astute critics will note that even the likelihood ratio is "contingent" in the sense that it is contingent on the parameterization of the models being compared; however this is a feature shared by all methods, Frequentist, Bayesian and Likelihoodist)
A: Decision theory is the underlying theory on which statistics operates.  The problem is to find a good (in some sense) procedure for producing decisions from data.  However, there's rarely an unambiguous choice of procedure, in the sense of minimizing expected loss, so other criteria must be invoked to choose among them.  Choosing the procedures that is Bayes with respect to some prior is one of these criteria, but it may not always be what you want.  Minimax might be more important in some case, or unbiasedness.
Anyone who insists that the frequentists are wrong or the Bayesians or wrong is mostly revealing their ignorance of statistics.
A: I'm going to give you an answer. Four drawbacks actually. Note that none of these are actually objections that should drive one all the way to frequentist analysis, but there are cons to going with a Bayesian framework:


*

*Choice of prior. This is the usual carping for a reason, though in my case it's not the usual "priors are subjective!" but that coming up with a prior that's well reasoned and actually represents your best attempt at summarizing a prior is a great deal of work in many cases. An entire aim of my dissertation, for example, can be summed up as "estimate priors".

*It's computationally intensive. Especially for models involving many variables. For a large dataset with many variables being estimated, it may very  well be prohibitively computationally intensive, especially in certain circumstances where the data cannot readily be thrown onto a cluster or the like. Some of the ways to resolve this, like augmented data rather than MCMC, are somewhat theoretically challenging, at least to me.

*Posterior distributions are somewhat more difficult to incorporate into a meta-analysis, unless a frequentist, parametric description of the distribution has been provided.

*Depending on what journal the analysis is intended for, either the use of Bayes generally, or your choice of priors, gives your paper slightly more points where a reviewer can dig into it. Some of these are reasonable reviewer objections, but some just stem from the nature of Bayes and how familiar people in some fields are with it.


None of these things should stop you. Indeed, none of these things have stopped me, and hopefully doing Bayesian analysis will help address at least number 4.
A: For some time I have wanted to educate myself more on Bayesian approaches to modeling to get past my cursory understanding (I have coded Gibbs samplers in graduate course work, but have never done anything real). Along the way though I have thought some of Brian Dennis' papers have been though-provoking and have wished I could find a Bayesian friend (the ones who weren't in the closet) to read the papers and hear their counterpoints. So, here are the papers I am referring to, but the quote I always remember is 

Being Bayesian means never having to say you're wrong.

http://faculty.washington.edu/skalski/classes/QERM597/papers/Dennis_1996.pdf
http://classes.warnercnr.colostate.edu/nr575/files/2011/01/Lele-and-Dennis-2009.pdf
A: What are the open problems in Bayesian Statistics from the ISBA quarterly newsletter list 5 problems with bayesian stats from various leaders in the field, #1 being, boringly enough, "Model selection and hypothesis testing".
A: I am a Bayesian by inclination, but generally a frequentist in practice.  The reason for this is usually that performing the full Bayesian analysis properly (rather than e.g. MAP solutions) for the types of problem I am interested in is tricky and computationally intensive.  Often a full Bayesian analysis is required to really see the benefit of this approach over frequentist equivalents.
For me, the trade-off is basically a choice between Bayesian methods that are conceptually elegant and easy to understand, but difficult to implement in practice and frequentist methods, which are conceptually awkward and subtle (try explaining how to interpret a hypothesis test accurately or why there isn't a 95% probability that the true value lies in a 95% confidence interval), but which are well suited to easily implemented "cookbook" solutions.
Horses for courses.
A: From a purely practical point of view, I am not a fan of methods which require lots of computation (I am thinking of Gibbs sampler and MCMC, often used in the Bayesian framework, but this also applies to e.g. bootstrap techniques in frequentist analysis). The reason being that any kind of debugging (testing the implementation, looking at robustness with respect to assumptions, etc) itself requires a bunch of Monte Carlo simulations, and you are quickly in a computational morass. I prefer the underlying analysis techniques to be fast and deterministic, even if they are only approximate. 
This is a purely practical objection, of course: given infinite computing resources, this objection would disappear. And it only applies to a subset of Bayesian methods. Also this is more of a preference given my workflow.
