# MCMC in a frequentist setting

I have been trying to get a sense of the different problems in frequentist settings where MCMC is used. I am familiar that MCMC (or Monte Carlo) is used in fitting GLMMs and in maybe Monte Carlo EM algorithms. Are there more frequentist problems where MCMC is used?

• When a Bayesian model can also be construed as a frequentist model (e.g., all priors are flat), the posterior mode is the MLE. So, you can use MCMC to do MLE, although that might not be a very good way to do it. – Kodiologist Sep 24 '16 at 6:41
• @Kodiologist Sure. Although, it is likely we are interested in the posterior mean (if working under least squares loss function), so we won't even try to find the MLE. But I see what you mean. – Greenparker Sep 24 '16 at 11:32
• @Kodiologist but why would frequentist do that? First, this would lead to multiple conceptual problems (assuming that parameter is r.v., how to interpret HDI's etc.). Second, if frequentist would use it simply instead of optimization algorithm to find point estimate, why would he do so since it's a very inefficient way if you are only after point estimate... – Tim Oct 2 '16 at 17:29
• I came across this by accident, but thought this a useful topic. If I'm not mistaken, monte carlo methods are generally concerned with sampling from a target distribution that one may, or may not, be able to sample directly from. The shift between Bayesian & Frequentist is interpretation of data as RVs or paramters are RVs (as stated by @Tim). So it seems to me that MC methods are neither "bayesian" nor "frequentist". It's rather the philosophy that is applied to their use that creates a distinction. Would this be a correct assessment? – Jon Jun 5 '17 at 16:15
• @Greenparker, so in classical MC, you may have the situation $E[h(x)] = \int h(x) f(x) dx \approx \frac{1}{n}\sum^n x_i$ where $x_i \sim f$ and $f$ is some instrumental distribution. If I am following your logic, the (direct) sampling from $f$ is neither bayesian nor frequentist, but the use of the empirical mean would then make this a frequentist estimator. Is this interpretation of your logic correct? – Jon Jun 5 '17 at 17:42

4. frequentist goodness-of-fit tests, which may require computations of coverage probabilities, $p$_values, powers, for sufficient or insufficient statistics with no closed-form density, or conditional on ancillary statistics. Take the example of testing for independence in (large) contingency tables (or deriving the maximum likelihood estimator).