I believe MCMC could be utilized to estimate the MAP. At least there is an option in packages like PyMC.
I just started reading about Bayesian Optimization, but the first thing that hit me was that it incrementally samples points from a function, much like MCMC.
Even if the nature of the function varies -- BO: black box and expensive versus MCMC: known distribution. I can image that BO can as well be utilized for a known distribution (maybe this where I am not clear yet).
My question is: how do BO and MCMC compare when it comes to MAP estimation, and in sampling efficiency in general? Specifically, is BO more efficient for MAP estimation than MCMC?