Sample a probability distribution with an evolutionary algorithm?

I've been doing some initial level reading on Markov chain Monte Carlo (MCMC).

For what I can tell given a probability distribution $P(x_1, x_2, ..., x_N)$ (dependent on $N$ parameters), MCMC algorithms can be used to both solve the optimization problem (ie, find $max(P)$ or $min(P)$) and at the same time sample $P$.

Thus we can estimate the uncertainty of the optimized parameters that resulted in $max(P)$ or $min(P)$, through the sampled distribution.

Evolutionary algorithms (I use a genetic algorithm) are mainly used to solve optimization problems, but apparently can not be used to infer uncertainties on the optimized parameters using the sampled distribution.

Is this correct? If so why? Both methods seem to rely on exploring the parameter space of the probability distribution by means of a MC process, coupled with some criteria to move forward in the exploration.

• I do not think MCMC algorithms can be used to "estimate the uncertainty of the optimized parameters that resulted in max(P) or min(P), through the sampled distribution"... Sep 14 '17 at 16:47
• I'm likely not expressing myself properly here. This is what I mean: dan.iel.fm/emcee/current/user/line/… Sep 14 '17 at 17:04