I have sometimes seen some statisticians used bayesian analysis and related techniques such as MCMC simply as a tool when a frequentist approach is not satisfying, typically for example when the maximum likelihood estimator is hard to find or takes too much time to compute.
In these case they focus only on the definition of the model and estimation (by MCMC for example) and barely on the choice prior distributions (from what I've seen).
I guess they use flat distributions by default but the particular choice of a prior is never discussed, maybe because of the large number of parameters the model may have and the difficulty to assign a meaningful prior distribution to each one of them.
Is it something usual to switch to a bayesian analysis and use it merely a as computational counterpart of a frequentist approach when the latter performs poorly?
My feeling is that bayesian and frequentist are very distinct statistical framework (at least that's what my teachers taught me!) and using one or another shouldn't be justified simply by computational purposes and moreover the choice of the prior distributions should at least be carefully made.
I know this is not really a statistical question but I just would like to know what pure bayesians think about this kind of use of bayesian analysis. Sorry in advance if it does not fit this site.
Best Answer
The set of methods called "frequentist" statistics is quite broad. It allows you to specify any proposed estimator you want and then investigate its long-run properties conditional on the true values of the parameters. This method only counts an estimator out completely if it is "inadmissible", meaning that it is dominated by another available estimator (i.e., it gives equal/higher risk over every possible value of the parameter and higher risk over at least some parameter values).
Now, there is a famous theorem that says that, under wide conditions, Bayesian estimators are admissible --- i.e., they are not dominated by other estimators. Bayesian estimators tend to be biased (since they incorporate prior information) but they are also consistent under fairly wide conditions. This means that they are estimators that will tend to perform well in terms of the frequentist criteria. Consequently, frequentists usually consider these estimators as one option that can be used in their analysis.
By definition, "pure Bayesians" are going to adopt the Bayesian methodology in all cases. Most pure Bayesians are going to have adopted this methodology by being convinced of its underlying philosophical and mathematical superiority. However, part of the motivation for adoption of Bayesian methods may be the knowledge that even under the frequentist paradigm, these methods tend to perform well according to frequentist criteria. As to what a pure Bayesian would think of a frequentist using a Bayesian estimator, I suppose it is somewhat like what a priest would think of an atheist who decides one day to pray for spiritual guidance (e.g., on the basis that it can't do any harm even under their own philosophy). They would likely see this as a desirable change in behaviour, improperly motivated, but also possibly a useful entry-point to try to convince them that the general philosophy underpinning that activity is coherent and desirable.