In the statement for estimating parameters through the Maximum Likelihood Principle (MLE), there is no mention of whether to choose a local maximum or a global maximum. (In my very limited reading so far) From the examples given in various textbooks/lecture notes, it seems that we should choose the global maximum of the likelihood function for inference. Is this correct?
The reason I am asking is because I am dealing with some data whose likelihood seems to have several maxima. The parameter space is three dimensional, so I have no intuition about the situation. In this case how do I estimate the parameters properly - do I just look for the maximum in a small part of the parameter space? (The bounds could be established through guesses based on the data, for example.)