Bayesian parameter estimation typically trains an emulator to reproduce the objective function using a limited number of design point (order 10 per dimension).

Bayesian parameter estimation typically trains an emulator to reproduce the objective function using a limited number of design point (order 10 per dimension). Once the emulator is trained, you could of course use a multi dimensional minimization function of your choice to find the best fit point. However, constructing and sampling the Bayesian posterior using MCMC methods has several advantages. Sometimes you can have a local minimum which is essentially flat, so the optimal hyperparameter is unstable. You'll see this in the posterior distribution. Or you could have two parameters which are correlated so it's their sum that's constrained not their individual values. All this information provides important context when understanding your model's uncertainty.