# 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

As indicated in the many comments, Markov Chain Monte Carlo is a special case of the Monte Carlo method, which is designed to approximate quantities related with a distribution via pseudo-random number simulation. As such, it has no connection with a particular statistical paradigm and the earliest instances of the method, as in Metropolis et al. (1953), were unrelated with statistics, Bayesian or frequentist. If anything these methods are naturally "frequentist" (an ill-defined category anyway) in that they rely on the stabilisation of the frequencies or averages towards the expectation as the number of simulations increases, aka the Law of Large Numbers.

It is therefore possible within non-Bayesian complex problems to use MCMC methods to replace intractable integrals. Check for instance

1. the optimisation of likelihoods with no closed form expressions, as in latent variable and random effect models. The EM algorithm may fail to work because of an intractable "E" step, in which case the expectation need be replaced by a Monte Carlo or a Markov Chain Monte Carlo approximation. With a possible evaluation of the error. Or it may fail to work because of an intractable "M" step, in which case the maximisation can sometimes be replaced by a Markovian maximisation procedure as in simulated annealing. Or using Gibbs steps.
2. simulated inference methods in econometrics, as the simulated method of moments, indirect inference, empirical likelihood.
3. approximations of likelihoods with intractable normalising constants such as Ising, Potts, and other Markov random fields models, using for instance exchange algorithms.
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).
5. again in econometrics, Laplace type estimators, "which include means and quantiles of quasi-posterior distributions defined as transformations of general nonlikelihood-based statistical criterion functions, such as those in GMM, nonlinear IV, empirical likelihood, and minimum distance methods" (Chernozhukov and Hong, 2003), rely on MCMC algorithms.