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.

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

The MCMC algorithms are used to simulate from a given target distribution and not to be confused with stochastic optimisation algorithms such as the simulated annealing algorithm or the Robbins-Monro algorithm. Both groups of algorithms used Markov chains to reach their goal, but the theories behind them differ.

I suggest you read some introduction to the topic, either on-line or in a textbook.

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