How can we make sure that Dynare actually finds the global max rather than a local max when it’s finding the mode of the posterior distribution?
I think Dynare takes the mean values of the prior as initial points to search for the mode of the posterior. Can I change the initial values without changing the prior distribution itself? If not, how can we check if the mode we get is actually a global max?
Suppose the program actually converged to a local max for the mode. I think Dynare starts the MH algorithm from the mode. Would the posterior distribution nevertheless converge to the true distribution, but I just need to have longer chains?
ideally, you should be able to traverse the whole distribution starting from anywhere in the posterior, not necessarily the mode. but i think that works less than perfectly in practice. one can look at the mh mat files and plot the posterior obtained from these files to check if the optimizer indeed found the ‘global’ max.
so in principle it’s ok that the program found a local max rather than global max as the mode since the posterior distributions obtained should be identical.
The dynare manual states that when the posterior is plotted alongside with the mode and the prior, if the mode is not at the mode of the posterior, there could be a problem. As I understand it, the problem is the mode found initially was not a global max. Right?
Nevertheless, the posterior distribution and its moments are still reliable given I used a long enough chain. Am I more or less correct in my understanding?
the point is that what is right in principle may not work in practice. the posterior simulator could get stuck in one part of the parameter space and not move from there at all. as far as i understand, the mode is your ‘updated’ MLE estimate, so the ‘best’ result your data gives you.
from my practical experience, i note that if the mode is not the ‘right’ one, one will need ‘abnormal’ values for the step-size of the mh simulator to get an acceptance rate of say 30%.
it is better to use mcmc and the optimizer simultaneously to find the right mode. the ‘global’ maximum is a valuable statistic and then mcmc simulates the posterior at least in the vicinity of the mode.