MCMC with flat prior vs. glmer Is MCMC-based mixed model with flat prior basically just a robust variant of a classical mixed model? I mean – frequentist analyses work with a flat prior anyway so the only difference should be in the more reliable method.
Am I right?
If so, what is the point in NOT using MCMC all the time on modern devices?
Thanks.
 A: The standard estimate for a frequentist model would be the mode vs. other estimates (mean, median) more common in Bayesian inference. That is one difference. Note also that it matters what parametrization you choose (e.g. hierarchical scale, log-scale, variance, precision etc. when it comes to these estimates).
The posterior may also be seriously hard to sample and standard Gibbs samplers may not be able to tell you that did not truly sample the posterior. Depending on what model we are talking about, you could even have an improper posterior, but happily sample away and not notice it (as has happened in a couple of published papers).
Runtime is also still as issue with Bayesian models on sufficiently large datasets (e.g. 50,000+ observations), particularly if you implement/parametrize them unwisely.
Another thought is that one should perhaps go a step further and consider always using proper weakly informative priors. For Gelman has argued for this point of view and is very much worth reading. 
