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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?

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The samples from the posterior allow you to compute expectations of the parameters. From Betancourt's A Conceptual Introduction to Hamiltonian Monte Carlo...

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

$$\hat{f}_{N}=\frac{1}{N} \sum_{n=0}^{N} f\left(q_{n}\right)$$

So if you want to estimate parameters for, say, a model then you have to compute the expectation of the samples for that parameter.

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    $\begingroup$ You don't have to compute expectations to form an estimate. For example, you might be interested in a maximum a posteriori (MAP) estimate, in which case you'd use an estimate of the maximum density region of the generated samples, or in an estimate which minimizes a custom loss function, as in inventory control. $\endgroup$
    – jbowman
    Jan 28, 2020 at 0:02
  • $\begingroup$ @jbowman the reason I use MCMC sampling is that the posterior is a very complicateed distribution. Can I still estimate these parameters without computing the posterior via MAP? Thanks for your anwser! $\endgroup$
    – independentvariable
    Jan 28, 2020 at 0:43
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    $\begingroup$ @jbowman That is true, but there are good reasons to use expectations over MAP (as outlined in that paper). In any case, you are right that "estimate" depends on what you want to know about the posterior, which is not necessarily the expectation. $\endgroup$
    – Demetri Pananos
    Jan 28, 2020 at 0:56
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    $\begingroup$ @independentvariable - MAP estimation would use the output of the MCMC sampler as an input - but I meant it as an example of an estimator of the parameters other than the average of the output samples, not as a blanket recommendation. $\endgroup$
    – jbowman
    Jan 28, 2020 at 1:42

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