What is the difference between Maximum Likelihood and Optimization or Constrained Maximum Likelihood and Constrained Optimization?
A. Optimization and Maximum Likelihood are nearly identical except that Maximum Likelihood has been designed to handle data sets. This allows it to provide four types of statistical inference – Wald, Profile Likelihood, Bootstrap, and Bayesian. It also allows for an additional descent method, the BHHH. Optimization could be used for maximum likelihood problems. However, since it doesn’t know about data sets, you would have to write your own procedures for statistical inference. Moreover, if your data sets were large and didn’t fit into memory, you would have to include code in your function for reading in the data. Maximum Likelihood does this for you. If your optimization problems did not involve maximum likelihood, you would be better off with Optimization because the additional features in Maximum Likelihood for handling data would just get in your way. But if you are solving maximum likelihood problems, Maximum Likelihood would save you the considerable effort required for handling
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