I'm still relatively new to understanding the bayesian mentality.
MCMC (e.g metropolis hasting) finds out the posterior distribution of the parameters of interest. MCMC requires taking many samples from the posterior distribution and creating a histogram.
the MAP estimate can be used to select a value of this parameter from the posterior distribution that well summarizes the posterior distribution. The MAP estimate can be expressed as the argmax(prior * likelihood).
Can "argmax(prior * likelihood)" be calculated using any optimization algorithm? For example, the Genetic Algorithm or Gradient Descent? For example, if we wanted to estimate the "lambda" parameter from an exponential distribution, could the Genetic Algorithm be used to evaluate the "argmax(prior * likelihood)" and identify the final lambda value?
It seems like MAP estimate is used to identify a final value of the parameter, whereas the posterior distribution of the parameter generated by MCMC is used to create a "credible interval" around the MAP estimate of the parameter?