This should be an easy question. So, I was reading an article with title "tree inference for single-cell data". I got how MCMC (Markov chain Monte Carlo) works. Then it comes the "after convergence" part which is a bit confusing. """After convergence, the MCMC chain can be used to sample trees and error rates proportionally to the joint posterior distribution in Eq.4. In addition, it is possible to obtain a single best fitting combination of mutation tree and error rates via point estimates of the model parameters. One way of doing this is via maximum a posteriori (MAP) estimates. $(T,\theta)_{MAP} = arg max_{(T,\theta)}P(T,\theta | D)$. Another possibility is to use ML estimates, i.e., $(T, \sigma, \theta)_{ML} = arg max_{(T, \sigma, \theta)} P(D|T,\sigma,\theta)$. """ I thought the motivation for using MCMC, is because the posterior probability is difficult to calculate with. After the chain convergences, the tree T and the error rates can be sampled easily. Even if we get the tree, we still cannot calculate the posterior probability. Then how to proceed with MAP to find the pair of tree T and error rates, which would maximum the posterior probability？For the ML estimation, it actually calculates the likelihood. Under the setting in this article, the likelihood is just a multiplication of a bunch of error rates. How is this ML approach connected with the MCMC sampling then？I will dive into their code of course. It would be great if someone would help to clarify a bit.
Following this link, https://stats.stackexchange.com/questions/168837/why-is-mcmc-needed-when-estimating-a-parameter-using-map I understand a bit more. It goes as the following. """ MAP refers to finding parameter values that (locally) maximize a posterior. It doesn't matter how one gets those parameter values: solving for maxima analytically, using a numerical routine, automatic differentiation, etc. """. and """ MAP estimation only requires us to be able to evaluate the (unnormalized) posterior distribution at a given 𝜃, and only gives us one piece of information about that posterior. By approximately sampling an intractable posterior, MCMC methods allow us to learn all information about the posterior distribution.""". But still confused about how to proceed. I would dive into the code to learn more.