Parameter Estimation via MCMC

In general, we use MCMC method to sample from a distribution which is hard to compute. In Bayesian setting, we sample from the posterior distribution of the random parameters defining the underlying distributions via MCMC.

My question is, by using MCMC, how can we estimate the parameters? What are the methods out there?

Given sufficient time, the history of the Markov chain,$$\{q_0,...,q_N\}$$, denoted samples generated by the Markov chain, becomes a convenient quantification of the typical set.In particular, we can estimate expectations across the typical set, and hence expectations across the entire parameter space, by averaging the target function over this history
$$\hat{f}_{N}=\frac{1}{N} \sum_{n=0}^{N} f\left(q_{n}\right)$$