# Monte Carlo maximum likelihood vs Bayesian inference

I recently heard about MCMLE (Monte Carlo maximum likelihood estimation) for finding $$\hat\theta = \underset{\theta}{\text{argmax}} \frac{\exp\left(\theta^TT(y)\right)}{c(\theta)}$$ when the normalization constant $$c(\theta)$$ is too hard to compute. The main reference I see is "Constrained Monte Carlo Maximum Likelihood for Dependent Data" by Geyer and Thompson (1992). In practice I've only encountered this with estimating ERGMs (exponential random graph models) but I gather it can be used more widely.

Briefly, the idea is that we can pick some $$\theta_0$$ and sample $$y_1,\dots,y_n$$ from $$p(y|\theta_0)$$ via MCMC (since $$c(\theta_0)$$ cancels out from the accept/reject ratios) and then we can use the fact that $$\frac 1n \sum_i e^{(\theta-\theta_0)^TT(y_i)} \to_p E_{\theta_0}(e^{(\theta-\theta_0)^TT(Y)}) = \frac{c(\theta)}{c(\theta_0)}$$ and this allows us to find the MLE. Generally this process requires running a full MCMC chain at every update to the current MLE solution.

My question: why would we do MLE at all here? This seems like a really weird in-between where we're going to go to all the effort of running MCMC chains but we're still insisting on frequentist inference. Why not just do a fully Bayesian analysis in this case? I know MLE is convenient and all but this just feels like an odd attempt to keep using a frequentist method when it's more work than just using the existing Bayesian tools. Or are there advantages I'm not appreciating?

• ah I just went through what the Metropolis acceptance ratio would actually be and i see i was making a mistake. If I'm trying to sample from $\pi(\theta|y) \propto p(y|\theta)\pi(\theta)$ I'm considering $y$ as fixed so the marginal likelihoods cancel but I'd still have to deal with something of the form $c(\theta) / c(\theta')$ so this actually isn't more work. Thanks! – alfalfa Oct 17 '18 at 18:39