Estimate MLE with Metropolis-Hastings algorithm

I am learning now about the Metropolis-Hastings algorithm. I want to understand better how we can use and apply this algorithm.
For example, consider a Poisson regression model.

So the probability mass function is given by $$p(y | x;\theta) = \frac{e^{-\lambda } \lambda^y}{y!}$$ where

$$\lambda := E[y|x] = e^{\theta x}.$$

If I am correct, we can approximate the maximum likelihood estimator for $\theta$ with a Monte Carlo procedure like the Metropolis-Hastings algorithm, right?

My question is: how can we do this? What will be the proposal distribution $q(x,y)$? What do we know if we have the transition and acceptance probabilities? How does this help to find the MLE for $\theta$?

I am not looking for a complete solution, but more a description of how we can approximate the MLE for this model.

• Metropolis-Hastings is not typically used to find maximum likelihood estimates - are you sure you are asking about that? – Juho Kokkala Nov 1 '17 at 6:14