There have been many debates within statistics between Bayesians and frequentists. I generally find these rather off-putting (although I think it has died down). On the other hand, I've met several people who take an entirely pragmatic view of the issue, saying that sometimes it is more convenient to conduct a frequentist analysis and sometimes it's easier to run a Bayesian analysis. I find this perspective practical and refreshing.
It occurs to me that it would be helpful to have a list of such cases. Because there are too many statistical analyses, and because I assume that it is ordinarily more practical to conduct a frequentist analysis (coding a t-test in WinBUGS is considerably more involved than the single function call required to perform the frequentist-based version in R, for example), it would be nice to have a list of the situations where a Bayesian approach is simpler, more practical, and / or more convenient than a frequentist approach.
(Two answers that I have no interest in are: 'always', and 'never'. I understand people have strong opinions, but please don't air them here. If this thread becomes a venue for petty squabbling, I will probably delete it. My goal here is to develop a resource that will be useful for an analyst with a job to do, not an axe to grind.)
People are welcome to suggest more than one case, but please use separate answers to do so, so that each situation can be evaluated (voted / discussed) individually. Answers should list: (1) what the nature of the situation is, and (2) why the Bayesian approach is simpler in this case. Some code (say, in WinBUGS) demonstrating how the analysis would be done and why the Bayesian version is more practical would be ideal, but I expect will be too cumbersome. If it can be done easily I would appreciate it, but please include why either way.
Finally, I recognize that I have not defined what it means for one approach to be 'simpler' than another. The truth is, I'm not entirely sure what it should mean for one approach to be more practical than the other. I'm open to different suggestions, just specify your interpretation when you explain why a Bayesian analysis is more convenient in the situation you discuss.
Best Answer
(1) In contexts where the likelihood function is intractable (at least numerically), the use of the Bayesian approach, by means of Approximate Bayesian Computation (ABC), has gained ground over some frequentist competitors such as composite likelihoods (1, 2) or the empirical likelihood because it tends to be easier to implement (not necessarily correct). Due to this, the use of ABC has become popular in areas where it is common to come across intractable likelihoods such as biology, genetics, and ecology. Here, we could mention an ocean of examples.
Some examples of intractable likelihoods are
Superposed processes. Cox and Smith (1954) proposed a model in the context of neurophysiology which consists of $N$ superposed point processes. For example consider the times between the electrical pulses observed at some part of the brain that were emited by several neurones during a certain period. This sample contains non iid observations which makes difficult to construct the corresponding likelihood, complicating the estimation of the corresponding parameters. A (partial)frequentist solution was recently proposed in this paper. The implementation of the ABC approach has also been recently studied and it can be found here.
Population genetics is another example of models leading to intractable likelihoods. In this case the intractability has a different nature: the likelihood is expressed in terms of a multidimensional integral (sometimes of dimension $1000+$) which would take a couple of decades just to evaluate it at a single point. This area is probably ABC's headquarters.